1951:
2661:
1266:
633:
1692:
785:
438:-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications. This is because of the relationship between a imprimitive character
417:
2489:
2398:
1782:
122:
1401:
1078:
1519:
959:
491:
1553:
3028:(1837). "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält".
2157:
2088:
2267:
1058:
1330:
656:
1929:
1855:
2213:
295:
483:
1820:
1010:
2656:{\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}={\frac {1}{k^{s}}}\sum _{r=1}^{k}\chi (r)\operatorname {\zeta } \left(s,{\frac {r}{k}}\right).}
846:
456:
145:
2314:
211:
1703:
1261:{\displaystyle L(s,\chi )=W(\chi )2^{s}\pi ^{s-1}q^{1/2-s}\sin \left({\frac {\pi }{2}}(s+\delta )\right)\Gamma (1-s)L(1-s,{\overline {\chi }}).}
37:
1337:
3131:
1424:
862:
628:{\displaystyle \chi (n)={\begin{cases}\chi ^{\star }(n),&\mathrm {if} \gcd(n,q)=1\\0,&\mathrm {if} \gcd(n,q)\neq 1\end{cases}}}
2976:
1687:{\displaystyle \Lambda (s,\chi )=q^{s/2}\pi ^{-(s+\delta )/2}\operatorname {\Gamma } \left({\frac {s+\delta }{2}}\right)L(s,\chi ).}
3328:
977:, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of
3058:
3070:
2985:
2941:
3313:
2116:
3175:
2055:
3292:
2218:
3358:
3087:. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society.
3015:
2889:
3256:
3124:
1939:
3054:
2873:
1015:
3297:
3282:
3025:
2687:
2672:
2278:
780:{\displaystyle L(s,\chi )=L(s,\chi ^{\star })\prod _{p\,|\,q}\left(1-{\frac {\chi ^{\star }(p)}{p^{s}}}\right)}
203:
2170:) ≤ 1, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line Re(
1281:
3389:
3318:
3102:
3042:
2881:
2692:
3222:
3092:
1884:
3117:
3097:
412:{\displaystyle L(s,\chi )=\prod _{p}\left(1-\chi (p)p^{-s}\right)^{-1}{\text{ for }}{\text{Re}}(s)>1,}
1825:
2177:
2273:
is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if
270:
515:
461:
1971:
1790:
980:
3208:
3183:
824:
3287:
3323:
3161:
2416:
853:
646:.) An application of the Euler product gives a simple relationship between the corresponding
168:
3251:
2995:
2951:
441:
286:
172:
130:
2959:
2899:
8:
974:
148:
2296:) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet
3277:
3231:
2682:
1994:
3065:. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press.
3343:
3333:
3066:
3011:
2981:
2971:
2937:
2885:
2393:{\displaystyle \beta <1-{\frac {c}{\log \!\!\;{\big (}q(2+|\gamma |){\big )}}}\ }
3261:
3213:
3003:
2955:
2895:
164:
156:
2878:
Ten lectures on the interface between analytic number theory and harmonic analysis
2991:
2947:
1858:
254:
794:, by analytic continuation, even though the Euler product is only valid when Re(
3109:
3080:
2929:
2113:) < 0 are simple zeros at −1, −3, −5, .... These correspond to the poles of
1272:
3383:
3192:
3147:
2281:
is the conjecture that all the non-trivial zeros lie on the critical line Re(
278:
176:
2936:, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag,
1777:{\displaystyle \Lambda (s,\chi )=W(\chi )\Lambda (1-s,{\overline {\chi }}).}
3368:
3363:
2880:. Regional Conference Series in Mathematics. Vol. 84. Providence, RI:
423:
214:
that also bears his name. In the course of the proof, Dirichlet shows that
117:{\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.}
2289:
242:
20:
3140:
2677:
2048:) < 0 are simple zeros at −2, −4, −6, .... (There is also a zero at
1950:
1396:{\displaystyle W(\chi )={\frac {\tau (\chi )}{i^{\delta }{\sqrt {q}}}}}
282:
1415:
160:
2974:; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
1514:{\displaystyle \tau (\chi )=\sum _{a=1}^{q}\chi (a)\exp(2\pi ia/q).}
2467:
has analytic properties that are closely related to the
Dirichlet
954:{\displaystyle L(s,\chi _{0})=\zeta (s)\prod _{p\,|\,q}(1-p^{-s})}
2018:
2967:
2431:
are linear combinations, with constant coefficients, of the
1547:
Another way to state the functional equation is in terms of
621:
1970:) = 1 − 3 + 5 − 7 + ⋅⋅⋅ (sometimes given the special name
2463:. This means that the Hurwitz zeta function for rational
2415:-functions may be written as a linear combination of the
2406:
1072:> 1. One way to express the functional equation is:
2812:
2810:
2808:
2806:
2804:
2120:
2059:
2492:
2317:
2292:, zero-free regions including and beyond the line Re(
2221:
2180:
2166:
The remaining zeros lie in the critical strip 0 ≤ Re(
2119:
2058:
1887:
1828:
1793:
1706:
1556:
1427:
1340:
1284:
1081:
1018:
983:
865:
827:
659:
494:
464:
444:
298:
133:
40:
2841:
2839:
2837:
2822:
2152:{\displaystyle \textstyle \Gamma ({\frac {s+1}{2}})}
2801:
810:-function of the primitive character which induces
2655:
2392:
2261:
2207:
2151:
2083:{\displaystyle \textstyle \Gamma ({\frac {s}{2}})}
2082:
1974:), with trivial zeros at the negative odd integers
1923:
1849:
1814:
1776:
1686:
1513:
1395:
1324:
1260:
1052:
1004:
953:
840:
779:
627:
477:
450:
411:
139:
116:
3063:Multiplicative number theory. I. Classical theory
3053:
2834:
2828:
2816:
2795:
2759:
2340:
2339:
814:, multiplied by only a finite number of factors.
3381:
3139:
3047:A Classical Introduction to Modern Number Theory
2262:{\displaystyle L(1-{\overline {\rho }},\chi )=0}
594:
550:
3079:
2856:
2854:
2845:
237:is principal, then the corresponding Dirichlet
2017:) < 0, there are zeros at certain negative
3125:
2379:
2344:
1697:The functional equation can be expressed as:
3040:
2851:
2783:
2269:too, because of the functional equation. If
212:theorem on primes in arithmetic progressions
3132:
3118:
2872:
2341:
1053:{\displaystyle L(1-s,{\overline {\chi }})}
163:greater than 1. It is a special case of a
3024:
3002:
2860:
2747:
2735:
2723:
920:
914:
720:
714:
207:
1949:
1325:{\displaystyle \chi (-1)=(-1)^{\delta }}
2977:NIST Handbook of Mathematical Functions
2965:
2928:
2912:
2771:
2711:
2052:= 0.) These correspond to the poles of
429:
3382:
2934:Introduction to analytic number theory
2419:at rational values. Fixing an integer
964:
798:) > 1.) The formula shows that the
3113:
2407:Relation to the Hurwitz zeta function
1787:The functional equation implies that
1524:It is a property of Gauss sums that |
821:-function of the principal character
3314:Birch and Swinnerton-Dyer conjecture
2163:These are called the trivial zeros.
1924:{\displaystyle L(s,\chi )=\zeta (s)}
277:-function can also be written as an
2479:. Then we can write its Dirichlet
13:
2530:
2288:Up to the possible existence of a
2121:
2097:(−1) = −1, then the only zeros of
2060:
1829:
1740:
1707:
1630:
1557:
1206:
590:
587:
546:
543:
78:
14:
3401:
3359:Main conjecture of Iwasawa theory
2427:-functions for characters modulo
1850:{\displaystyle \Lambda (s,\chi )}
852:can be expressed in terms of the
3049:(2nd ed.). Springer-Verlag.
2304:a non-real character of modulus
2208:{\displaystyle L(\rho ,\chi )=0}
1982:be a primitive character modulo
1940:Functional equation (L-function)
1271:In this equation, Γ denotes the
1064:be a primitive character modulo
260:
202:These functions are named after
2906:
2866:
3293:Ramanujan–Petersson conjecture
3283:Generalized Riemann hypothesis
3179:-functions of Hecke characters
2980:, Cambridge University Press,
2789:
2777:
2765:
2753:
2741:
2729:
2717:
2705:
2673:Generalized Riemann hypothesis
2613:
2607:
2547:
2541:
2508:
2496:
2471:-functions. Specifically, let
2374:
2370:
2362:
2352:
2279:generalized Riemann hypothesis
2250:
2225:
2196:
2184:
2145:
2124:
2076:
2063:
1918:
1912:
1903:
1891:
1869:is primitive character modulo
1844:
1832:
1809:
1797:
1768:
1743:
1737:
1731:
1722:
1710:
1678:
1666:
1616:
1604:
1572:
1560:
1505:
1482:
1473:
1467:
1437:
1431:
1368:
1362:
1350:
1344:
1313:
1303:
1297:
1288:
1252:
1227:
1221:
1209:
1198:
1186:
1112:
1106:
1097:
1085:
1047:
1022:
999:
987:
948:
926:
916:
903:
897:
888:
869:
756:
750:
716:
703:
684:
675:
663:
609:
597:
565:
553:
534:
528:
504:
498:
478:{\displaystyle \chi ^{\star }}
422:where the product is over all
397:
391:
351:
345:
314:
302:
204:Peter Gustav Lejeune Dirichlet
95:
89:
56:
44:
1:
3252:Analytic class number formula
3083:; Kowalski, Emmanuel (2004).
2922:
2882:American Mathematical Society
2829:Montgomery & Vaughan 2006
2817:Montgomery & Vaughan 2006
2796:Montgomery & Vaughan 2006
2760:Montgomery & Vaughan 2006
2693:Special values of L-functions
2300:-functions: for example, for
16:Type of mathematical function
3257:Riemann–von Mangoldt formula
3008:Multiplicative Number Theory
2403:for β + iγ a non-real zero.
2277:is a complex character. The
2239:
2032:(−1) = 1, the only zeros of
1865:. (Again, this assumes that
1763:
1247:
1042:
790:(This formula holds for all
458:and the primitive character
265:Since a Dirichlet character
7:
3098:Encyclopedia of Mathematics
2846:Iwaniec & Kowalski 2004
2666:
10:
3406:
3010:(3rd ed.). Springer.
1938:For generalizations, see:
1815:{\displaystyle L(s,\chi )}
1005:{\displaystyle L(s,\chi )}
171:, it can be extended to a
31:is a function of the form
3342:
3306:
3270:
3244:
3201:
3154:
2750:, chapter 5, equation (3)
2738:, chapter 5, equation (2)
841:{\displaystyle \chi _{0}}
271:completely multiplicative
3030:Abhand. Ak. Wiss. Berlin
2784:Ireland & Rosen 1990
2698:
1945:
206:who introduced them in (
3209:Dedekind zeta functions
2966:Apostol, T. M. (2010),
2786:, chapter 16, section 4
1972:Dirichlet beta function
817:As a special case, the
179:, and is then called a
3093:"Dirichlet-L-function"
3085:Analytic Number Theory
2968:"Dirichlet L-function"
2657:
2603:
2534:
2475:be a character modulo
2394:
2263:
2209:
2153:
2084:
1975:
1925:
1851:
1816:
1778:
1688:
1515:
1463:
1397:
1326:
1262:
1054:
1006:
955:
842:
781:
629:
479:
452:
413:
141:
118:
82:
3329:Bloch–Kato conjecture
3324:Beilinson conjectures
3307:Algebraic conjectures
3162:Riemann zeta function
2658:
2583:
2514:
2417:Hurwitz zeta function
2395:
2264:
2210:
2174:) = 1/2. That is, if
2154:
2085:
1953:
1926:
1852:
1817:
1779:
1689:
1516:
1443:
1398:
1327:
1263:
1055:
1007:
973:-functions satisfy a
956:
854:Riemann zeta function
843:
782:
630:
480:
453:
451:{\displaystyle \chi }
414:
169:analytic continuation
142:
140:{\displaystyle \chi }
119:
62:
3390:Zeta and L-functions
3334:Langlands conjecture
3319:Deligne's conjecture
3271:Analytic conjectures
2490:
2315:
2219:
2178:
2117:
2056:
1885:
1826:
1791:
1704:
1554:
1425:
1338:
1282:
1079:
1016:
981:
863:
825:
657:
492:
462:
442:
430:Primitive characters
296:
287:absolute convergence
249:= 1. Otherwise, the
173:meromorphic function
131:
38:
3288:Lindelöf hypothesis
3055:Montgomery, Hugh L.
3026:Dirichlet, P. G. L.
2874:Montgomery, Hugh L.
2423:≥ 1, the Dirichlet
975:functional equation
965:Functional equation
149:Dirichlet character
3278:Riemann hypothesis
3202:Algebraic examples
3059:Vaughan, Robert C.
3041:Ireland, Kenneth;
2972:Olver, Frank W. J.
2683:Modularity theorem
2653:
2390:
2259:
2205:
2149:
2148:
2080:
2079:
1976:
1921:
1847:
1812:
1774:
1684:
1511:
1393:
1322:
1258:
1050:
1002:
951:
925:
838:
777:
725:
642:is the modulus of
625:
620:
485:which induces it:
475:
448:
409:
329:
233:= 1. Moreover, if
137:
114:
3377:
3376:
3155:Analytic examples
3072:978-0-521-84903-6
2987:978-0-521-19225-5
2943:978-0-387-90163-3
2643:
2581:
2561:
2389:
2385:
2242:
2143:
2074:
2013:) > 1. For Re(
1766:
1657:
1391:
1388:
1250:
1184:
1045:
906:
770:
706:
389:
384:
320:
187:and also denoted
109:
3397:
3298:Artin conjecture
3262:Weil conjectures
3134:
3127:
3120:
3111:
3110:
3106:
3088:
3076:
3050:
3037:
3021:
2998:
2962:
2916:
2910:
2904:
2903:
2870:
2864:
2858:
2849:
2843:
2832:
2826:
2820:
2814:
2799:
2793:
2787:
2781:
2775:
2769:
2763:
2757:
2751:
2745:
2739:
2733:
2727:
2721:
2715:
2709:
2688:Artin conjecture
2662:
2660:
2659:
2654:
2649:
2645:
2644:
2636:
2620:
2602:
2597:
2582:
2580:
2579:
2567:
2562:
2560:
2559:
2550:
2536:
2533:
2528:
2399:
2397:
2396:
2391:
2387:
2386:
2384:
2383:
2382:
2373:
2365:
2348:
2347:
2331:
2268:
2266:
2265:
2260:
2243:
2235:
2214:
2212:
2211:
2206:
2158:
2156:
2155:
2150:
2144:
2139:
2128:
2089:
2087:
2086:
2081:
2075:
2067:
1930:
1928:
1927:
1922:
1859:entire functions
1856:
1854:
1853:
1848:
1821:
1819:
1818:
1813:
1783:
1781:
1780:
1775:
1767:
1759:
1693:
1691:
1690:
1685:
1662:
1658:
1653:
1642:
1633:
1628:
1627:
1623:
1595:
1594:
1590:
1520:
1518:
1517:
1512:
1501:
1462:
1457:
1402:
1400:
1399:
1394:
1392:
1390:
1389:
1384:
1382:
1381:
1371:
1357:
1331:
1329:
1328:
1323:
1321:
1320:
1267:
1265:
1264:
1259:
1251:
1243:
1205:
1201:
1185:
1177:
1164:
1163:
1153:
1140:
1139:
1124:
1123:
1059:
1057:
1056:
1051:
1046:
1038:
1012:to the value of
1011:
1009:
1008:
1003:
960:
958:
957:
952:
947:
946:
924:
919:
887:
886:
847:
845:
844:
839:
837:
836:
806:is equal to the
786:
784:
783:
778:
776:
772:
771:
769:
768:
759:
749:
748:
738:
724:
719:
702:
701:
634:
632:
631:
626:
624:
623:
593:
549:
527:
526:
484:
482:
481:
476:
474:
473:
457:
455:
454:
449:
418:
416:
415:
410:
390:
387:
385:
382:
380:
379:
371:
367:
366:
365:
328:
241:-function has a
228:
165:Dirichlet series
157:complex variable
146:
144:
143:
138:
123:
121:
120:
115:
110:
108:
107:
98:
84:
81:
76:
3405:
3404:
3400:
3399:
3398:
3396:
3395:
3394:
3380:
3379:
3378:
3373:
3338:
3302:
3266:
3240:
3197:
3150:
3138:
3091:
3081:Iwaniec, Henryk
3073:
3018:
2988:
2944:
2930:Apostol, Tom M.
2925:
2920:
2919:
2911:
2907:
2892:
2884:. p. 163.
2871:
2867:
2859:
2852:
2844:
2835:
2827:
2823:
2815:
2802:
2794:
2790:
2782:
2778:
2770:
2766:
2758:
2754:
2746:
2742:
2734:
2730:
2722:
2718:
2710:
2706:
2701:
2669:
2635:
2628:
2624:
2616:
2598:
2587:
2575:
2571:
2566:
2555:
2551:
2537:
2535:
2529:
2518:
2491:
2488:
2487:
2409:
2378:
2377:
2369:
2361:
2343:
2342:
2335:
2330:
2316:
2313:
2312:
2234:
2220:
2217:
2216:
2179:
2176:
2175:
2129:
2127:
2118:
2115:
2114:
2066:
2057:
2054:
2053:
1948:
1886:
1883:
1882:
1827:
1824:
1823:
1792:
1789:
1788:
1758:
1705:
1702:
1701:
1643:
1641:
1637:
1629:
1619:
1600:
1596:
1586:
1582:
1578:
1555:
1552:
1551:
1544:) | = 1.
1540: (
1528: (
1497:
1458:
1447:
1426:
1423:
1422:
1410: (
1383:
1377:
1373:
1372:
1358:
1356:
1339:
1336:
1335:
1316:
1312:
1283:
1280:
1279:
1242:
1176:
1175:
1171:
1149:
1145:
1141:
1129:
1125:
1119:
1115:
1080:
1077:
1076:
1037:
1017:
1014:
1013:
982:
979:
978:
967:
939:
935:
915:
910:
882:
878:
864:
861:
860:
832:
828:
826:
823:
822:
764:
760:
744:
740:
739:
737:
730:
726:
715:
710:
697:
693:
658:
655:
654:
619:
618:
586:
584:
575:
574:
542:
540:
522:
518:
511:
510:
493:
490:
489:
469:
465:
463:
460:
459:
443:
440:
439:
432:
386:
383: for
381:
372:
358:
354:
335:
331:
330:
324:
297:
294:
293:
263:
229:is non-zero at
215:
210:) to prove the
132:
129:
128:
103:
99:
85:
83:
77:
66:
39:
36:
35:
17:
12:
11:
5:
3403:
3393:
3392:
3375:
3374:
3372:
3371:
3366:
3361:
3355:
3353:
3340:
3339:
3337:
3336:
3331:
3326:
3321:
3316:
3310:
3308:
3304:
3303:
3301:
3300:
3295:
3290:
3285:
3280:
3274:
3272:
3268:
3267:
3265:
3264:
3259:
3254:
3248:
3246:
3242:
3241:
3239:
3238:
3229:
3220:
3211:
3205:
3203:
3199:
3198:
3196:
3195:
3190:
3181:
3173:
3164:
3158:
3156:
3152:
3151:
3137:
3136:
3129:
3122:
3114:
3108:
3107:
3089:
3077:
3071:
3051:
3043:Rosen, Michael
3038:
3022:
3016:
3000:
2986:
2963:
2942:
2924:
2921:
2918:
2917:
2905:
2890:
2865:
2861:Davenport 2000
2850:
2833:
2821:
2800:
2788:
2776:
2764:
2752:
2748:Davenport 2000
2740:
2736:Davenport 2000
2728:
2724:Davenport 2000
2716:
2714:, Theorem 11.7
2703:
2702:
2700:
2697:
2696:
2695:
2690:
2685:
2680:
2675:
2668:
2665:
2664:
2663:
2652:
2648:
2642:
2639:
2634:
2631:
2627:
2623:
2619:
2615:
2612:
2609:
2606:
2601:
2596:
2593:
2590:
2586:
2578:
2574:
2570:
2565:
2558:
2554:
2549:
2546:
2543:
2540:
2532:
2527:
2524:
2521:
2517:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2483:-function as:
2411:The Dirichlet
2408:
2405:
2401:
2400:
2381:
2376:
2372:
2368:
2364:
2360:
2357:
2354:
2351:
2346:
2338:
2334:
2329:
2326:
2323:
2320:
2258:
2255:
2252:
2249:
2246:
2241:
2238:
2233:
2230:
2227:
2224:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2161:
2160:
2147:
2142:
2138:
2135:
2132:
2126:
2123:
2091:
2078:
2073:
2070:
2065:
2062:
1954:The Dirichlet
1947:
1944:
1931:has a pole at
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1846:
1843:
1840:
1837:
1834:
1831:
1811:
1808:
1805:
1802:
1799:
1796:
1785:
1784:
1773:
1770:
1765:
1762:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1695:
1694:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1661:
1656:
1652:
1649:
1646:
1640:
1636:
1632:
1626:
1622:
1618:
1615:
1612:
1609:
1606:
1603:
1599:
1593:
1589:
1585:
1581:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1522:
1521:
1510:
1507:
1504:
1500:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1461:
1456:
1453:
1450:
1446:
1442:
1439:
1436:
1433:
1430:
1404:
1403:
1387:
1380:
1376:
1370:
1367:
1364:
1361:
1355:
1352:
1349:
1346:
1343:
1333:
1319:
1315:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1273:gamma function
1269:
1268:
1257:
1254:
1249:
1246:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1204:
1200:
1197:
1194:
1191:
1188:
1183:
1180:
1174:
1170:
1167:
1162:
1159:
1156:
1152:
1148:
1144:
1138:
1135:
1132:
1128:
1122:
1118:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1049:
1044:
1041:
1036:
1033:
1030:
1027:
1024:
1021:
1001:
998:
995:
992:
989:
986:
966:
963:
962:
961:
950:
945:
942:
938:
934:
931:
928:
923:
918:
913:
909:
905:
902:
899:
896:
893:
890:
885:
881:
877:
874:
871:
868:
835:
831:
788:
787:
775:
767:
763:
758:
755:
752:
747:
743:
736:
733:
729:
723:
718:
713:
709:
705:
700:
696:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
636:
635:
622:
617:
614:
611:
608:
605:
602:
599:
596:
592:
589:
585:
583:
580:
577:
576:
573:
570:
567:
564:
561:
558:
555:
552:
548:
545:
541:
539:
536:
533:
530:
525:
521:
517:
516:
514:
509:
506:
503:
500:
497:
472:
468:
447:
434:Results about
431:
428:
420:
419:
408:
405:
402:
399:
396:
393:
378:
375:
370:
364:
361:
357:
353:
350:
347:
344:
341:
338:
334:
327:
323:
319:
316:
313:
310:
307:
304:
301:
262:
259:
208:Dirichlet 1837
136:
125:
124:
113:
106:
102:
97:
94:
91:
88:
80:
75:
72:
69:
65:
61:
58:
55:
52:
49:
46:
43:
15:
9:
6:
4:
3:
2:
3402:
3391:
3388:
3387:
3385:
3370:
3367:
3365:
3362:
3360:
3357:
3356:
3354:
3352:
3350:
3346:
3341:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3311:
3309:
3305:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3275:
3273:
3269:
3263:
3260:
3258:
3255:
3253:
3250:
3249:
3247:
3243:
3237:
3235:
3230:
3228:
3226:
3221:
3219:
3217:
3212:
3210:
3207:
3206:
3204:
3200:
3194:
3193:Selberg class
3191:
3189:
3187:
3182:
3180:
3178:
3174:
3172:
3170:
3165:
3163:
3160:
3159:
3157:
3153:
3149:
3148:number theory
3145:
3143:
3135:
3130:
3128:
3123:
3121:
3116:
3115:
3112:
3104:
3100:
3099:
3094:
3090:
3086:
3082:
3078:
3074:
3068:
3064:
3060:
3056:
3052:
3048:
3044:
3039:
3035:
3031:
3027:
3023:
3019:
3017:0-387-95097-4
3013:
3009:
3005:
3004:Davenport, H.
3001:
2997:
2993:
2989:
2983:
2979:
2978:
2973:
2969:
2964:
2961:
2957:
2953:
2949:
2945:
2939:
2935:
2931:
2927:
2926:
2915:, p. 249
2914:
2909:
2901:
2897:
2893:
2891:0-8218-0737-4
2887:
2883:
2879:
2875:
2869:
2862:
2857:
2855:
2847:
2842:
2840:
2838:
2831:, p. 332
2830:
2825:
2819:, p. 333
2818:
2813:
2811:
2809:
2807:
2805:
2798:, p. 121
2797:
2792:
2785:
2780:
2774:, p. 262
2773:
2768:
2762:, p. 282
2761:
2756:
2749:
2744:
2737:
2732:
2725:
2720:
2713:
2708:
2704:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2670:
2650:
2646:
2640:
2637:
2632:
2629:
2625:
2621:
2617:
2610:
2604:
2599:
2594:
2591:
2588:
2584:
2576:
2572:
2568:
2563:
2556:
2552:
2544:
2538:
2525:
2522:
2519:
2515:
2511:
2505:
2502:
2499:
2493:
2486:
2485:
2484:
2482:
2478:
2474:
2470:
2466:
2462:
2459:= 1, 2, ...,
2458:
2454:
2450:
2446:
2442:
2438:
2434:
2430:
2426:
2422:
2418:
2414:
2404:
2366:
2358:
2355:
2349:
2336:
2332:
2327:
2324:
2321:
2318:
2311:
2310:
2309:
2307:
2303:
2299:
2295:
2291:
2286:
2284:
2280:
2276:
2272:
2256:
2253:
2247:
2244:
2236:
2231:
2228:
2222:
2202:
2199:
2193:
2190:
2187:
2181:
2173:
2169:
2164:
2140:
2136:
2133:
2130:
2112:
2108:
2104:
2100:
2096:
2092:
2071:
2068:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2026:
2025:
2023:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1993:There are no
1991:
1989:
1985:
1981:
1973:
1969:
1965:
1961:
1957:
1952:
1943:
1941:
1936:
1934:
1915:
1909:
1906:
1900:
1897:
1894:
1888:
1880:
1876:
1872:
1868:
1864:
1860:
1841:
1838:
1835:
1806:
1803:
1800:
1794:
1771:
1760:
1755:
1752:
1749:
1746:
1734:
1728:
1725:
1719:
1716:
1713:
1700:
1699:
1698:
1681:
1675:
1672:
1669:
1663:
1659:
1654:
1650:
1647:
1644:
1638:
1634:
1624:
1620:
1613:
1610:
1607:
1601:
1597:
1591:
1587:
1583:
1579:
1575:
1569:
1566:
1563:
1550:
1549:
1548:
1545:
1543:
1539:
1535:
1531:
1527:
1508:
1502:
1498:
1494:
1491:
1488:
1485:
1479:
1476:
1470:
1464:
1459:
1454:
1451:
1448:
1444:
1440:
1434:
1428:
1421:
1420:
1419:
1417:
1413:
1409:
1385:
1378:
1374:
1365:
1359:
1353:
1347:
1341:
1334:
1317:
1309:
1306:
1300:
1294:
1291:
1285:
1278:
1277:
1276:
1274:
1255:
1244:
1239:
1236:
1233:
1230:
1224:
1218:
1215:
1212:
1202:
1195:
1192:
1189:
1181:
1178:
1172:
1168:
1165:
1160:
1157:
1154:
1150:
1146:
1142:
1136:
1133:
1130:
1126:
1120:
1116:
1109:
1103:
1100:
1094:
1091:
1088:
1082:
1075:
1074:
1073:
1071:
1067:
1063:
1039:
1034:
1031:
1028:
1025:
1019:
996:
993:
990:
984:
976:
972:
943:
940:
936:
932:
929:
921:
911:
907:
900:
894:
891:
883:
879:
875:
872:
866:
859:
858:
857:
855:
851:
833:
829:
820:
815:
813:
809:
805:
802:-function of
801:
797:
793:
773:
765:
761:
753:
745:
741:
734:
731:
727:
721:
711:
707:
698:
694:
690:
687:
681:
678:
672:
669:
666:
660:
653:
652:
651:
649:
645:
641:
615:
612:
606:
603:
600:
581:
578:
571:
568:
562:
559:
556:
537:
531:
523:
519:
512:
507:
501:
495:
488:
487:
486:
470:
466:
445:
437:
427:
425:
424:prime numbers
406:
403:
400:
394:
376:
373:
368:
362:
359:
355:
348:
342:
339:
336:
332:
325:
321:
317:
311:
308:
305:
299:
292:
291:
290:
288:
284:
280:
279:Euler product
276:
272:
268:
261:Euler product
258:
256:
253:-function is
252:
248:
244:
240:
236:
232:
226:
222:
218:
213:
209:
205:
200:
198:
194:
190:
186:
184:
178:
177:complex plane
175:on the whole
174:
170:
166:
162:
158:
154:
150:
134:
111:
104:
100:
92:
86:
73:
70:
67:
63:
59:
53:
50:
47:
41:
34:
33:
32:
30:
28:
22:
3369:Euler system
3364:Selmer group
3348:
3344:
3233:
3224:
3215:
3185:
3184:Automorphic
3176:
3168:
3166:
3141:
3096:
3084:
3062:
3046:
3033:
3029:
3007:
2975:
2933:
2913:Apostol 1976
2908:
2877:
2868:
2848:, p. 84
2824:
2791:
2779:
2772:Apostol 1976
2767:
2755:
2743:
2731:
2719:
2712:Apostol 1976
2707:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2440:
2436:
2432:
2428:
2424:
2420:
2412:
2410:
2402:
2305:
2301:
2297:
2293:
2287:
2282:
2274:
2270:
2171:
2167:
2165:
2162:
2110:
2106:
2102:
2098:
2094:
2049:
2045:
2041:
2037:
2033:
2029:
2021:
2014:
2010:
2006:
2002:
1998:
1992:
1987:
1983:
1979:
1977:
1967:
1963:
1959:
1955:
1937:
1932:
1878:
1874:
1870:
1866:
1862:
1786:
1696:
1546:
1541:
1537:
1533:
1532:) | =
1529:
1525:
1523:
1411:
1407:
1405:
1270:
1069:
1065:
1061:
970:
968:
849:
818:
816:
811:
807:
803:
799:
795:
791:
789:
650:-functions:
647:
643:
639:
637:
435:
433:
421:
274:
266:
264:
250:
246:
238:
234:
230:
224:
220:
216:
201:
196:
192:
188:
182:
180:
152:
126:
26:
24:
18:
3223:Hasse–Weil
2863:, chapter 9
2726:, chapter 5
2290:Siegel zero
1877:> 1. If
1332: ; and
243:simple pole
21:mathematics
3351:-functions
3236:-functions
3227:-functions
3218:-functions
3188:-functions
3171:-functions
3167:Dirichlet
3144:-functions
2960:0335.10001
2923:References
2900:0814.11001
2678:L-function
2308:, we have
2109:) with Re(
2044:) with Re(
2009:) with Re(
1958:-function
1881:= 1, then
969:Dirichlet
283:half-plane
181:Dirichlet
25:Dirichlet
3103:EMS Press
2622:
2618:ζ
2605:χ
2585:∑
2539:χ
2531:∞
2516:∑
2506:χ
2367:γ
2328:−
2319:β
2285:) = 1/2.
2248:χ
2240:¯
2237:ρ
2232:−
2194:χ
2188:ρ
2122:Γ
2061:Γ
1910:ζ
1901:χ
1842:χ
1830:Λ
1807:χ
1764:¯
1761:χ
1750:−
1741:Λ
1735:χ
1720:χ
1708:Λ
1676:χ
1651:δ
1635:
1631:Γ
1614:δ
1602:−
1598:π
1570:χ
1558:Λ
1489:π
1480:
1465:χ
1445:∑
1435:χ
1429:τ
1416:Gauss sum
1379:δ
1366:χ
1360:τ
1348:χ
1318:δ
1307:−
1292:−
1286:χ
1248:¯
1245:χ
1234:−
1216:−
1207:Γ
1196:δ
1179:π
1169:
1158:−
1134:−
1127:π
1110:χ
1095:χ
1043:¯
1040:χ
1029:−
997:χ
941:−
933:−
908:∏
895:ζ
880:χ
830:χ
746:⋆
742:χ
735:−
708:∏
699:⋆
695:χ
673:χ
613:≠
524:⋆
520:χ
496:χ
471:⋆
467:χ
446:χ
374:−
360:−
343:χ
340:−
322:∏
312:χ
185:-function
161:real part
135:χ
87:χ
79:∞
64:∑
54:χ
3384:Category
3245:Theorems
3232:Motivic
3061:(2006).
3045:(1990).
3006:(2000).
2932:(1976),
2876:(1994).
2667:See also
2443:) where
2019:integers
1990:> 1.
1068:, where
3105:, 2001
2996:2723248
2952:0434929
1986:, with
1414:) is a
848:modulo
638:(Here,
281:in the
29:-series
3347:-adic
3214:Artin
3069:
3014:
2994:
2984:
2958:
2950:
2940:
2898:
2888:
2473:χ
2388:
1935:= 1.)
1857:) are
1536:, so |
1406:where
1060:. Let
273:, its
255:entire
127:where
2970:, in
2699:Notes
2215:then
1995:zeros
1946:Zeros
1873:with
1822:(and
167:. By
159:with
147:is a
3067:ISBN
3012:ISBN
2982:ISBN
2938:ISBN
2886:ISBN
2455:and
2322:<
1978:Let
401:>
151:and
23:, a
3146:in
2956:Zbl
2896:Zbl
2337:log
2093:If
2028:If
1997:of
1861:of
1477:exp
1275:;
1166:sin
595:gcd
551:gcd
285:of
269:is
245:at
199:).
19:In
3386::
3101:,
3095:,
3057:;
3034:48
3032:.
2992:MR
2990:,
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