Dirichlet series
| L(s) = 1 | − 2-s + 5·3-s − 20·4-s + 19·5-s − 5·6-s + 51·7-s + 11·8-s − 64·9-s − 19·10-s + 27·11-s − 100·12-s + 15·13-s − 51·14-s + 95·15-s + 148·16-s + 82·17-s + 64·18-s − 78·19-s − 380·20-s + 255·21-s − 27·22-s + 61·23-s + 55·24-s − 369·25-s − 15·26-s − 439·27-s − 1.02e3·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.962·3-s − 5/2·4-s + 1.69·5-s − 0.340·6-s + 2.75·7-s + 0.486·8-s − 2.37·9-s − 0.600·10-s + 0.740·11-s − 2.40·12-s + 0.320·13-s − 0.973·14-s + 1.63·15-s + 2.31·16-s + 1.16·17-s + 0.838·18-s − 0.941·19-s − 4.24·20-s + 2.64·21-s − 0.261·22-s + 0.553·23-s + 0.467·24-s − 2.95·25-s − 0.113·26-s − 3.12·27-s − 6.88·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(43^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.38797\times 10^{20}\) |
| Root analytic conductor: | \(10.4448\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 43^{20} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.7088272241\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7088272241\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + T + 21 T^{2} + 15 p T^{3} + 291 T^{4} + 493 T^{5} + 3299 T^{6} + 1505 p^{2} T^{7} + 7565 p^{2} T^{8} + 1803 p^{5} T^{9} + 3947 p^{6} T^{10} + 1803 p^{8} T^{11} + 7565 p^{8} T^{12} + 1505 p^{11} T^{13} + 3299 p^{12} T^{14} + 493 p^{15} T^{15} + 291 p^{18} T^{16} + 15 p^{22} T^{17} + 21 p^{24} T^{18} + p^{27} T^{19} + p^{30} T^{20} \) |
| 3 | \( 1 - 5 T + 89 T^{2} - 326 T^{3} + 4555 T^{4} - 4681 p T^{5} + 192137 T^{6} - 556492 T^{7} + 6664441 T^{8} - 16945357 T^{9} + 192259862 T^{10} - 16945357 p^{3} T^{11} + 6664441 p^{6} T^{12} - 556492 p^{9} T^{13} + 192137 p^{12} T^{14} - 4681 p^{16} T^{15} + 4555 p^{18} T^{16} - 326 p^{21} T^{17} + 89 p^{24} T^{18} - 5 p^{27} T^{19} + p^{30} T^{20} \) | |
| 5 | \( 1 - 19 T + 146 p T^{2} - 2329 p T^{3} + 53018 p T^{4} - 3590441 T^{5} + 63812448 T^{6} - 751352097 T^{7} + 11330307869 T^{8} - 118903300014 T^{9} + 1578548875564 T^{10} - 118903300014 p^{3} T^{11} + 11330307869 p^{6} T^{12} - 751352097 p^{9} T^{13} + 63812448 p^{12} T^{14} - 3590441 p^{15} T^{15} + 53018 p^{19} T^{16} - 2329 p^{22} T^{17} + 146 p^{25} T^{18} - 19 p^{27} T^{19} + p^{30} T^{20} \) | |
| 7 | \( 1 - 51 T + 2958 T^{2} - 98283 T^{3} + 3538173 T^{4} - 93650638 T^{5} + 2648528462 T^{6} - 59561803592 T^{7} + 1408530194217 T^{8} - 27332735053531 T^{9} + 555667189834924 T^{10} - 27332735053531 p^{3} T^{11} + 1408530194217 p^{6} T^{12} - 59561803592 p^{9} T^{13} + 2648528462 p^{12} T^{14} - 93650638 p^{15} T^{15} + 3538173 p^{18} T^{16} - 98283 p^{21} T^{17} + 2958 p^{24} T^{18} - 51 p^{27} T^{19} + p^{30} T^{20} \) | |
| 11 | \( 1 - 27 T + 5077 T^{2} - 61984 T^{3} + 15043654 T^{4} - 162363748 T^{5} + 35128969454 T^{6} - 264446154846 T^{7} + 61397630693497 T^{8} - 405185649214743 T^{9} + 91897922249805306 T^{10} - 405185649214743 p^{3} T^{11} + 61397630693497 p^{6} T^{12} - 264446154846 p^{9} T^{13} + 35128969454 p^{12} T^{14} - 162363748 p^{15} T^{15} + 15043654 p^{18} T^{16} - 61984 p^{21} T^{17} + 5077 p^{24} T^{18} - 27 p^{27} T^{19} + p^{30} T^{20} \) | |
| 13 | \( 1 - 15 T + 16061 T^{2} - 162762 T^{3} + 119085587 T^{4} - 746573887 T^{5} + 549572425527 T^{6} - 1904744585888 T^{7} + 1797963317266135 T^{8} - 3466193660344645 T^{9} + 4474548970144863094 T^{10} - 3466193660344645 p^{3} T^{11} + 1797963317266135 p^{6} T^{12} - 1904744585888 p^{9} T^{13} + 549572425527 p^{12} T^{14} - 746573887 p^{15} T^{15} + 119085587 p^{18} T^{16} - 162762 p^{21} T^{17} + 16061 p^{24} T^{18} - 15 p^{27} T^{19} + p^{30} T^{20} \) | |
| 17 | \( 1 - 82 T + 31542 T^{2} - 2729726 T^{3} + 495433616 T^{4} - 43592811872 T^{5} + 5079597607931 T^{6} - 435855391738836 T^{7} + 37675035599844963 T^{8} - 176519231034637998 p T^{9} + \)\(21\!\cdots\!17\)\( T^{10} - 176519231034637998 p^{4} T^{11} + 37675035599844963 p^{6} T^{12} - 435855391738836 p^{9} T^{13} + 5079597607931 p^{12} T^{14} - 43592811872 p^{15} T^{15} + 495433616 p^{18} T^{16} - 2729726 p^{21} T^{17} + 31542 p^{24} T^{18} - 82 p^{27} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 + 78 T + 36391 T^{2} + 2240906 T^{3} + 592121360 T^{4} + 29939098218 T^{5} + 6238019086681 T^{6} + 15025921374898 p T^{7} + 52918040345784791 T^{8} + 2341393657746342360 T^{9} + \)\(39\!\cdots\!04\)\( T^{10} + 2341393657746342360 p^{3} T^{11} + 52918040345784791 p^{6} T^{12} + 15025921374898 p^{10} T^{13} + 6238019086681 p^{12} T^{14} + 29939098218 p^{15} T^{15} + 592121360 p^{18} T^{16} + 2240906 p^{21} T^{17} + 36391 p^{24} T^{18} + 78 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( 1 - 61 T + 85545 T^{2} - 5735688 T^{3} + 3578624869 T^{4} - 244193475031 T^{5} + 96258657044673 T^{6} - 6286223931296694 T^{7} + 1836703681458483033 T^{8} - \)\(10\!\cdots\!83\)\( T^{9} + \)\(25\!\cdots\!30\)\( T^{10} - \)\(10\!\cdots\!83\)\( p^{3} T^{11} + 1836703681458483033 p^{6} T^{12} - 6286223931296694 p^{9} T^{13} + 96258657044673 p^{12} T^{14} - 244193475031 p^{15} T^{15} + 3578624869 p^{18} T^{16} - 5735688 p^{21} T^{17} + 85545 p^{24} T^{18} - 61 p^{27} T^{19} + p^{30} T^{20} \) | |
| 29 | \( 1 - 53 T + 141138 T^{2} - 8023609 T^{3} + 9724245343 T^{4} - 610047550838 T^{5} + 443117344967692 T^{6} - 30265015746923862 T^{7} + 15138818067933540037 T^{8} - \)\(10\!\cdots\!31\)\( T^{9} + \)\(14\!\cdots\!70\)\( p T^{10} - \)\(10\!\cdots\!31\)\( p^{3} T^{11} + 15138818067933540037 p^{6} T^{12} - 30265015746923862 p^{9} T^{13} + 443117344967692 p^{12} T^{14} - 610047550838 p^{15} T^{15} + 9724245343 p^{18} T^{16} - 8023609 p^{21} T^{17} + 141138 p^{24} T^{18} - 53 p^{27} T^{19} + p^{30} T^{20} \) | |
| 31 | \( 1 + 253 T + 105884 T^{2} + 17011687 T^{3} + 6345862600 T^{4} + 759122832695 T^{5} + 238432130617916 T^{6} + 21648160854623909 T^{7} + 8562912517186200751 T^{8} + \)\(64\!\cdots\!12\)\( T^{9} + \)\(24\!\cdots\!36\)\( T^{10} + \)\(64\!\cdots\!12\)\( p^{3} T^{11} + 8562912517186200751 p^{6} T^{12} + 21648160854623909 p^{9} T^{13} + 238432130617916 p^{12} T^{14} + 759122832695 p^{15} T^{15} + 6345862600 p^{18} T^{16} + 17011687 p^{21} T^{17} + 105884 p^{24} T^{18} + 253 p^{27} T^{19} + p^{30} T^{20} \) | |
| 37 | \( 1 - 129 T + 285619 T^{2} - 40107546 T^{3} + 41430046973 T^{4} - 5853675673369 T^{5} + 4034937015391223 T^{6} - 546994538186490476 T^{7} + \)\(29\!\cdots\!87\)\( T^{8} - \)\(36\!\cdots\!35\)\( T^{9} + \)\(16\!\cdots\!36\)\( T^{10} - \)\(36\!\cdots\!35\)\( p^{3} T^{11} + \)\(29\!\cdots\!87\)\( p^{6} T^{12} - 546994538186490476 p^{9} T^{13} + 4034937015391223 p^{12} T^{14} - 5853675673369 p^{15} T^{15} + 41430046973 p^{18} T^{16} - 40107546 p^{21} T^{17} + 285619 p^{24} T^{18} - 129 p^{27} T^{19} + p^{30} T^{20} \) | |
| 41 | \( 1 - 391 T + 466858 T^{2} - 159315937 T^{3} + 107799629054 T^{4} - 32107569594125 T^{5} + 16028988094146372 T^{6} - 4181844238111783617 T^{7} + \)\(16\!\cdots\!81\)\( T^{8} - \)\(38\!\cdots\!70\)\( T^{9} + \)\(13\!\cdots\!96\)\( T^{10} - \)\(38\!\cdots\!70\)\( p^{3} T^{11} + \)\(16\!\cdots\!81\)\( p^{6} T^{12} - 4181844238111783617 p^{9} T^{13} + 16028988094146372 p^{12} T^{14} - 32107569594125 p^{15} T^{15} + 107799629054 p^{18} T^{16} - 159315937 p^{21} T^{17} + 466858 p^{24} T^{18} - 391 p^{27} T^{19} + p^{30} T^{20} \) | |
| 47 | \( 1 + 334 T + 469033 T^{2} + 2674182 p T^{3} + 97290890628 T^{4} + 20127660597266 T^{5} + 12996286592743642 T^{6} + 2230581953572455798 T^{7} + \)\(14\!\cdots\!19\)\( T^{8} + \)\(24\!\cdots\!68\)\( T^{9} + \)\(16\!\cdots\!34\)\( T^{10} + \)\(24\!\cdots\!68\)\( p^{3} T^{11} + \)\(14\!\cdots\!19\)\( p^{6} T^{12} + 2230581953572455798 p^{9} T^{13} + 12996286592743642 p^{12} T^{14} + 20127660597266 p^{15} T^{15} + 97290890628 p^{18} T^{16} + 2674182 p^{22} T^{17} + 469033 p^{24} T^{18} + 334 p^{27} T^{19} + p^{30} T^{20} \) | |
| 53 | \( 1 + 773 T + 1139233 T^{2} + 720257454 T^{3} + 606121338023 T^{4} + 324763897859933 T^{5} + 202760779719698827 T^{6} + 93491859975188097460 T^{7} + \)\(47\!\cdots\!43\)\( T^{8} + \)\(18\!\cdots\!79\)\( T^{9} + \)\(82\!\cdots\!10\)\( T^{10} + \)\(18\!\cdots\!79\)\( p^{3} T^{11} + \)\(47\!\cdots\!43\)\( p^{6} T^{12} + 93491859975188097460 p^{9} T^{13} + 202760779719698827 p^{12} T^{14} + 324763897859933 p^{15} T^{15} + 606121338023 p^{18} T^{16} + 720257454 p^{21} T^{17} + 1139233 p^{24} T^{18} + 773 p^{27} T^{19} + p^{30} T^{20} \) | |
| 59 | \( 1 + 1483 T + 1378157 T^{2} + 969130766 T^{3} + 686264212828 T^{4} + 451249739385536 T^{5} + 271319879713111662 T^{6} + \)\(14\!\cdots\!74\)\( T^{7} + \)\(73\!\cdots\!15\)\( T^{8} + \)\(36\!\cdots\!33\)\( T^{9} + \)\(17\!\cdots\!86\)\( T^{10} + \)\(36\!\cdots\!33\)\( p^{3} T^{11} + \)\(73\!\cdots\!15\)\( p^{6} T^{12} + \)\(14\!\cdots\!74\)\( p^{9} T^{13} + 271319879713111662 p^{12} T^{14} + 451249739385536 p^{15} T^{15} + 686264212828 p^{18} T^{16} + 969130766 p^{21} T^{17} + 1378157 p^{24} T^{18} + 1483 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 + 437 T + 935885 T^{2} + 416954406 T^{3} + 554190861587 T^{4} + 216661010493785 T^{5} + 3762278983936399 p T^{6} + 83061468261243730036 T^{7} + \)\(72\!\cdots\!59\)\( T^{8} + \)\(23\!\cdots\!99\)\( T^{9} + \)\(18\!\cdots\!58\)\( T^{10} + \)\(23\!\cdots\!99\)\( p^{3} T^{11} + \)\(72\!\cdots\!59\)\( p^{6} T^{12} + 83061468261243730036 p^{9} T^{13} + 3762278983936399 p^{13} T^{14} + 216661010493785 p^{15} T^{15} + 554190861587 p^{18} T^{16} + 416954406 p^{21} T^{17} + 935885 p^{24} T^{18} + 437 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 - 642 T + 2067386 T^{2} - 1339263368 T^{3} + 2018557242358 T^{4} - 1330913370861868 T^{5} + 1251783460266323431 T^{6} - \)\(82\!\cdots\!10\)\( T^{7} + \)\(55\!\cdots\!99\)\( T^{8} - \)\(34\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!89\)\( T^{10} - \)\(34\!\cdots\!00\)\( p^{3} T^{11} + \)\(55\!\cdots\!99\)\( p^{6} T^{12} - \)\(82\!\cdots\!10\)\( p^{9} T^{13} + 1251783460266323431 p^{12} T^{14} - 1330913370861868 p^{15} T^{15} + 2018557242358 p^{18} T^{16} - 1339263368 p^{21} T^{17} + 2067386 p^{24} T^{18} - 642 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( 1 - 1545 T + 2551637 T^{2} - 2382888830 T^{3} + 2318471332835 T^{4} - 1505208541247391 T^{5} + 1076364279273182533 T^{6} - \)\(50\!\cdots\!52\)\( T^{7} + \)\(44\!\cdots\!15\)\( p T^{8} - \)\(11\!\cdots\!05\)\( T^{9} + \)\(90\!\cdots\!10\)\( T^{10} - \)\(11\!\cdots\!05\)\( p^{3} T^{11} + \)\(44\!\cdots\!15\)\( p^{7} T^{12} - \)\(50\!\cdots\!52\)\( p^{9} T^{13} + 1076364279273182533 p^{12} T^{14} - 1505208541247391 p^{15} T^{15} + 2318471332835 p^{18} T^{16} - 2382888830 p^{21} T^{17} + 2551637 p^{24} T^{18} - 1545 p^{27} T^{19} + p^{30} T^{20} \) | |
| 73 | \( 1 + 1292 T + 2470024 T^{2} + 2540022046 T^{3} + 3113964531230 T^{4} + 2703184247613182 T^{5} + 2581291938334464755 T^{6} + \)\(19\!\cdots\!84\)\( T^{7} + \)\(15\!\cdots\!31\)\( T^{8} + \)\(10\!\cdots\!52\)\( T^{9} + \)\(68\!\cdots\!47\)\( T^{10} + \)\(10\!\cdots\!52\)\( p^{3} T^{11} + \)\(15\!\cdots\!31\)\( p^{6} T^{12} + \)\(19\!\cdots\!84\)\( p^{9} T^{13} + 2581291938334464755 p^{12} T^{14} + 2703184247613182 p^{15} T^{15} + 3113964531230 p^{18} T^{16} + 2540022046 p^{21} T^{17} + 2470024 p^{24} T^{18} + 1292 p^{27} T^{19} + p^{30} T^{20} \) | |
| 79 | \( 1 - 1405 T + 4124293 T^{2} - 4455739938 T^{3} + 7389320412515 T^{4} - 82764585783401 p T^{5} + 7995580193148085185 T^{6} - \)\(60\!\cdots\!84\)\( T^{7} + \)\(60\!\cdots\!69\)\( T^{8} - \)\(39\!\cdots\!21\)\( T^{9} + \)\(34\!\cdots\!50\)\( T^{10} - \)\(39\!\cdots\!21\)\( p^{3} T^{11} + \)\(60\!\cdots\!69\)\( p^{6} T^{12} - \)\(60\!\cdots\!84\)\( p^{9} T^{13} + 7995580193148085185 p^{12} T^{14} - 82764585783401 p^{16} T^{15} + 7389320412515 p^{18} T^{16} - 4455739938 p^{21} T^{17} + 4124293 p^{24} T^{18} - 1405 p^{27} T^{19} + p^{30} T^{20} \) | |
| 83 | \( 1 + 543 T + 3505821 T^{2} + 2026676938 T^{3} + 6084107600779 T^{4} + 3508580210105525 T^{5} + 6967823140297410865 T^{6} + \)\(38\!\cdots\!20\)\( T^{7} + \)\(58\!\cdots\!89\)\( T^{8} + \)\(29\!\cdots\!71\)\( T^{9} + \)\(37\!\cdots\!50\)\( T^{10} + \)\(29\!\cdots\!71\)\( p^{3} T^{11} + \)\(58\!\cdots\!89\)\( p^{6} T^{12} + \)\(38\!\cdots\!20\)\( p^{9} T^{13} + 6967823140297410865 p^{12} T^{14} + 3508580210105525 p^{15} T^{15} + 6084107600779 p^{18} T^{16} + 2026676938 p^{21} T^{17} + 3505821 p^{24} T^{18} + 543 p^{27} T^{19} + p^{30} T^{20} \) | |
| 89 | \( 1 - 2196 T + 7101672 T^{2} - 11169690002 T^{3} + 21314174237118 T^{4} - 26581378923914546 T^{5} + 37764446306913683967 T^{6} - \)\(39\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!27\)\( p T^{8} - \)\(38\!\cdots\!96\)\( T^{9} + \)\(37\!\cdots\!75\)\( T^{10} - \)\(38\!\cdots\!96\)\( p^{3} T^{11} + \)\(50\!\cdots\!27\)\( p^{7} T^{12} - \)\(39\!\cdots\!68\)\( p^{9} T^{13} + 37764446306913683967 p^{12} T^{14} - 26581378923914546 p^{15} T^{15} + 21314174237118 p^{18} T^{16} - 11169690002 p^{21} T^{17} + 7101672 p^{24} T^{18} - 2196 p^{27} T^{19} + p^{30} T^{20} \) | |
| 97 | \( 1 + 425 T + 4855100 T^{2} + 2674736785 T^{3} + 12413935569789 T^{4} + 7844309405033348 T^{5} + 21709474743257501488 T^{6} + \)\(14\!\cdots\!84\)\( T^{7} + \)\(28\!\cdots\!94\)\( T^{8} + \)\(18\!\cdots\!26\)\( T^{9} + \)\(29\!\cdots\!88\)\( T^{10} + \)\(18\!\cdots\!26\)\( p^{3} T^{11} + \)\(28\!\cdots\!94\)\( p^{6} T^{12} + \)\(14\!\cdots\!84\)\( p^{9} T^{13} + 21709474743257501488 p^{12} T^{14} + 7844309405033348 p^{15} T^{15} + 12413935569789 p^{18} T^{16} + 2674736785 p^{21} T^{17} + 4855100 p^{24} T^{18} + 425 p^{27} T^{19} + p^{30} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.97257608783465118137639171498, −2.79499945105508193691854945327, −2.73689782453096151427867968015, −2.48437655707580562663471495044, −2.46206471649445698373695748842, −2.33814924996146383500750085210, −2.06206924578734887506253768321, −2.00702574576325153081295416266, −1.97566901175042930322615625049, −1.95430622438171532173727540012, −1.75949474414354663646171602371, −1.70444129201573265146854798194, −1.61137144731644730067261363458, −1.55253354341545104181203428241, −1.50952441159867586096194939098, −1.36709396724507482687165728312, −1.18557503134810431402764930068, −0.823072372074985041512029491214, −0.810481481669711392338698472765, −0.65259950674283047476809505101, −0.58983717256422655862055226499, −0.47857319146709944974204688947, −0.27035579450452965897494216337, −0.14538442847082076064107796853, −0.07676076021099062498189183301, 0.07676076021099062498189183301, 0.14538442847082076064107796853, 0.27035579450452965897494216337, 0.47857319146709944974204688947, 0.58983717256422655862055226499, 0.65259950674283047476809505101, 0.810481481669711392338698472765, 0.823072372074985041512029491214, 1.18557503134810431402764930068, 1.36709396724507482687165728312, 1.50952441159867586096194939098, 1.55253354341545104181203428241, 1.61137144731644730067261363458, 1.70444129201573265146854798194, 1.75949474414354663646171602371, 1.95430622438171532173727540012, 1.97566901175042930322615625049, 2.00702574576325153081295416266, 2.06206924578734887506253768321, 2.33814924996146383500750085210, 2.46206471649445698373695748842, 2.48437655707580562663471495044, 2.73689782453096151427867968015, 2.79499945105508193691854945327, 2.97257608783465118137639171498
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.