Dirichlet series
| L(s) = 1 | + 12·2-s + 78·4-s − 3·5-s − 6·7-s + 364·8-s − 36·10-s − 14·11-s − 3·13-s − 72·14-s + 1.36e3·16-s − 8·17-s − 4·19-s − 234·20-s − 168·22-s − 13·23-s − 20·25-s − 36·26-s − 468·28-s − 23·29-s − 14·31-s + 4.36e3·32-s − 96·34-s + 18·35-s − 19·37-s − 48·38-s − 1.09e3·40-s − 30·41-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 39·4-s − 1.34·5-s − 2.26·7-s + 128.·8-s − 11.3·10-s − 4.22·11-s − 0.832·13-s − 19.2·14-s + 341.·16-s − 1.94·17-s − 0.917·19-s − 52.3·20-s − 35.8·22-s − 2.71·23-s − 4·25-s − 7.06·26-s − 88.4·28-s − 4.27·29-s − 2.51·31-s + 772.·32-s − 16.4·34-s + 3.04·35-s − 3.12·37-s − 7.78·38-s − 172.·40-s − 4.68·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{36} \cdot 149^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.94635\times 10^{21}\) |
| Root analytic conductor: | \(8.01546\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{36} \cdot 149^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{12} \) |
| 3 | \( 1 \) | |
| 149 | \( ( 1 - T )^{12} \) | |
| good | 5 | \( 1 + 3 T + 29 T^{2} + 83 T^{3} + 434 T^{4} + 1119 T^{5} + 4424 T^{6} + 10464 T^{7} + 34918 T^{8} + 15546 p T^{9} + 228728 T^{10} + 473919 T^{11} + 1254877 T^{12} + 473919 p T^{13} + 228728 p^{2} T^{14} + 15546 p^{4} T^{15} + 34918 p^{4} T^{16} + 10464 p^{5} T^{17} + 4424 p^{6} T^{18} + 1119 p^{7} T^{19} + 434 p^{8} T^{20} + 83 p^{9} T^{21} + 29 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + 6 T + 53 T^{2} + 242 T^{3} + 1308 T^{4} + 4846 T^{5} + 20325 T^{6} + 9224 p T^{7} + 230705 T^{8} + 653407 T^{9} + 2089904 T^{10} + 5407980 T^{11} + 2266141 p T^{12} + 5407980 p T^{13} + 2089904 p^{2} T^{14} + 653407 p^{3} T^{15} + 230705 p^{4} T^{16} + 9224 p^{6} T^{17} + 20325 p^{6} T^{18} + 4846 p^{7} T^{19} + 1308 p^{8} T^{20} + 242 p^{9} T^{21} + 53 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 14 T + 13 p T^{2} + 96 p T^{3} + 6775 T^{4} + 37077 T^{5} + 185806 T^{6} + 838016 T^{7} + 3549961 T^{8} + 13913987 T^{9} + 52145040 T^{10} + 183959742 T^{11} + 626990511 T^{12} + 183959742 p T^{13} + 52145040 p^{2} T^{14} + 13913987 p^{3} T^{15} + 3549961 p^{4} T^{16} + 838016 p^{5} T^{17} + 185806 p^{6} T^{18} + 37077 p^{7} T^{19} + 6775 p^{8} T^{20} + 96 p^{10} T^{21} + 13 p^{11} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 3 T + 55 T^{2} + 141 T^{3} + 1951 T^{4} + 4476 T^{5} + 50093 T^{6} + 103950 T^{7} + 1024442 T^{8} + 1954957 T^{9} + 17370533 T^{10} + 30386550 T^{11} + 245221481 T^{12} + 30386550 p T^{13} + 17370533 p^{2} T^{14} + 1954957 p^{3} T^{15} + 1024442 p^{4} T^{16} + 103950 p^{5} T^{17} + 50093 p^{6} T^{18} + 4476 p^{7} T^{19} + 1951 p^{8} T^{20} + 141 p^{9} T^{21} + 55 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 8 T + 129 T^{2} + 717 T^{3} + 6947 T^{4} + 28060 T^{5} + 211253 T^{6} + 567037 T^{7} + 3951482 T^{8} + 4417300 T^{9} + 48836470 T^{10} - 44174689 T^{11} + 2101235 p^{2} T^{12} - 44174689 p T^{13} + 48836470 p^{2} T^{14} + 4417300 p^{3} T^{15} + 3951482 p^{4} T^{16} + 567037 p^{5} T^{17} + 211253 p^{6} T^{18} + 28060 p^{7} T^{19} + 6947 p^{8} T^{20} + 717 p^{9} T^{21} + 129 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 4 T + 81 T^{2} + 258 T^{3} + 3843 T^{4} + 10693 T^{5} + 125306 T^{6} + 298422 T^{7} + 168345 p T^{8} + 6708439 T^{9} + 68590826 T^{10} + 131253174 T^{11} + 1346523227 T^{12} + 131253174 p T^{13} + 68590826 p^{2} T^{14} + 6708439 p^{3} T^{15} + 168345 p^{5} T^{16} + 298422 p^{5} T^{17} + 125306 p^{6} T^{18} + 10693 p^{7} T^{19} + 3843 p^{8} T^{20} + 258 p^{9} T^{21} + 81 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 13 T + 142 T^{2} + 1143 T^{3} + 8673 T^{4} + 59528 T^{5} + 394597 T^{6} + 2406983 T^{7} + 14176685 T^{8} + 77968405 T^{9} + 418881201 T^{10} + 4033417 p^{2} T^{11} + 10521531035 T^{12} + 4033417 p^{3} T^{13} + 418881201 p^{2} T^{14} + 77968405 p^{3} T^{15} + 14176685 p^{4} T^{16} + 2406983 p^{5} T^{17} + 394597 p^{6} T^{18} + 59528 p^{7} T^{19} + 8673 p^{8} T^{20} + 1143 p^{9} T^{21} + 142 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 23 T + 395 T^{2} + 4877 T^{3} + 51758 T^{4} + 469685 T^{5} + 3858600 T^{6} + 28750460 T^{7} + 199074860 T^{8} + 44365122 p T^{9} + 7851838688 T^{10} + 45444927621 T^{11} + 250591717289 T^{12} + 45444927621 p T^{13} + 7851838688 p^{2} T^{14} + 44365122 p^{4} T^{15} + 199074860 p^{4} T^{16} + 28750460 p^{5} T^{17} + 3858600 p^{6} T^{18} + 469685 p^{7} T^{19} + 51758 p^{8} T^{20} + 4877 p^{9} T^{21} + 395 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 14 T + 160 T^{2} + 1419 T^{3} + 12651 T^{4} + 93889 T^{5} + 681265 T^{6} + 4370044 T^{7} + 28168003 T^{8} + 165872107 T^{9} + 1001880791 T^{10} + 5534240029 T^{11} + 31712932293 T^{12} + 5534240029 p T^{13} + 1001880791 p^{2} T^{14} + 165872107 p^{3} T^{15} + 28168003 p^{4} T^{16} + 4370044 p^{5} T^{17} + 681265 p^{6} T^{18} + 93889 p^{7} T^{19} + 12651 p^{8} T^{20} + 1419 p^{9} T^{21} + 160 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 19 T + 435 T^{2} + 6117 T^{3} + 83873 T^{4} + 934343 T^{5} + 9736417 T^{6} + 89608258 T^{7} + 769486826 T^{8} + 6007577417 T^{9} + 44012372877 T^{10} + 296078381605 T^{11} + 1877232834877 T^{12} + 296078381605 p T^{13} + 44012372877 p^{2} T^{14} + 6007577417 p^{3} T^{15} + 769486826 p^{4} T^{16} + 89608258 p^{5} T^{17} + 9736417 p^{6} T^{18} + 934343 p^{7} T^{19} + 83873 p^{8} T^{20} + 6117 p^{9} T^{21} + 435 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 30 T + 630 T^{2} + 242 p T^{3} + 132934 T^{4} + 1536253 T^{5} + 15942916 T^{6} + 149623872 T^{7} + 1294936134 T^{8} + 10367083477 T^{9} + 77655276302 T^{10} + 544482255134 T^{11} + 3596802825854 T^{12} + 544482255134 p T^{13} + 77655276302 p^{2} T^{14} + 10367083477 p^{3} T^{15} + 1294936134 p^{4} T^{16} + 149623872 p^{5} T^{17} + 15942916 p^{6} T^{18} + 1536253 p^{7} T^{19} + 132934 p^{8} T^{20} + 242 p^{10} T^{21} + 630 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 15 T + 246 T^{2} + 2619 T^{3} + 30033 T^{4} + 285144 T^{5} + 2655040 T^{6} + 21909881 T^{7} + 177479979 T^{8} + 1346416517 T^{9} + 9873603109 T^{10} + 68255483876 T^{11} + 456026671185 T^{12} + 68255483876 p T^{13} + 9873603109 p^{2} T^{14} + 1346416517 p^{3} T^{15} + 177479979 p^{4} T^{16} + 21909881 p^{5} T^{17} + 2655040 p^{6} T^{18} + 285144 p^{7} T^{19} + 30033 p^{8} T^{20} + 2619 p^{9} T^{21} + 246 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - T + 258 T^{2} + 686 T^{3} + 32746 T^{4} + 173833 T^{5} + 3159377 T^{6} + 19666499 T^{7} + 250177506 T^{8} + 1522093066 T^{9} + 16078008221 T^{10} + 89911618542 T^{11} + 839979747933 T^{12} + 89911618542 p T^{13} + 16078008221 p^{2} T^{14} + 1522093066 p^{3} T^{15} + 250177506 p^{4} T^{16} + 19666499 p^{5} T^{17} + 3159377 p^{6} T^{18} + 173833 p^{7} T^{19} + 32746 p^{8} T^{20} + 686 p^{9} T^{21} + 258 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 16 T + 395 T^{2} + 4441 T^{3} + 67581 T^{4} + 623297 T^{5} + 7577923 T^{6} + 61778296 T^{7} + 651876542 T^{8} + 4808080523 T^{9} + 45326804659 T^{10} + 304835394648 T^{11} + 2617377641701 T^{12} + 304835394648 p T^{13} + 45326804659 p^{2} T^{14} + 4808080523 p^{3} T^{15} + 651876542 p^{4} T^{16} + 61778296 p^{5} T^{17} + 7577923 p^{6} T^{18} + 623297 p^{7} T^{19} + 67581 p^{8} T^{20} + 4441 p^{9} T^{21} + 395 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 26 T + 739 T^{2} + 13160 T^{3} + 227786 T^{4} + 3166215 T^{5} + 41699950 T^{6} + 479752210 T^{7} + 5194514199 T^{8} + 50976622899 T^{9} + 470037330734 T^{10} + 3994115219949 T^{11} + 31880627164561 T^{12} + 3994115219949 p T^{13} + 470037330734 p^{2} T^{14} + 50976622899 p^{3} T^{15} + 5194514199 p^{4} T^{16} + 479752210 p^{5} T^{17} + 41699950 p^{6} T^{18} + 3166215 p^{7} T^{19} + 227786 p^{8} T^{20} + 13160 p^{9} T^{21} + 739 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 16 T + 458 T^{2} + 6342 T^{3} + 102854 T^{4} + 1253623 T^{5} + 15283299 T^{6} + 164257682 T^{7} + 1687820982 T^{8} + 15974386386 T^{9} + 145384440141 T^{10} + 1216682940343 T^{11} + 9937155937675 T^{12} + 1216682940343 p T^{13} + 145384440141 p^{2} T^{14} + 15974386386 p^{3} T^{15} + 1687820982 p^{4} T^{16} + 164257682 p^{5} T^{17} + 15283299 p^{6} T^{18} + 1253623 p^{7} T^{19} + 102854 p^{8} T^{20} + 6342 p^{9} T^{21} + 458 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 39 T + 1249 T^{2} + 27317 T^{3} + 524434 T^{4} + 8280215 T^{5} + 118933121 T^{6} + 1498064992 T^{7} + 17521520034 T^{8} + 185053888185 T^{9} + 1834186332603 T^{10} + 16629487163152 T^{11} + 142069839742145 T^{12} + 16629487163152 p T^{13} + 1834186332603 p^{2} T^{14} + 185053888185 p^{3} T^{15} + 17521520034 p^{4} T^{16} + 1498064992 p^{5} T^{17} + 118933121 p^{6} T^{18} + 8280215 p^{7} T^{19} + 524434 p^{8} T^{20} + 27317 p^{9} T^{21} + 1249 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 15 T + 575 T^{2} + 6821 T^{3} + 150303 T^{4} + 1537962 T^{5} + 25378542 T^{6} + 235061520 T^{7} + 3179437558 T^{8} + 26943753841 T^{9} + 311687255347 T^{10} + 2405044514488 T^{11} + 24550756948835 T^{12} + 2405044514488 p T^{13} + 311687255347 p^{2} T^{14} + 26943753841 p^{3} T^{15} + 3179437558 p^{4} T^{16} + 235061520 p^{5} T^{17} + 25378542 p^{6} T^{18} + 1537962 p^{7} T^{19} + 150303 p^{8} T^{20} + 6821 p^{9} T^{21} + 575 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 2 T + 430 T^{2} + 1781 T^{3} + 97769 T^{4} + 544940 T^{5} + 15643484 T^{6} + 97962235 T^{7} + 1933240123 T^{8} + 12369386893 T^{9} + 190995008857 T^{10} + 1175619536159 T^{11} + 15381529083285 T^{12} + 1175619536159 p T^{13} + 190995008857 p^{2} T^{14} + 12369386893 p^{3} T^{15} + 1933240123 p^{4} T^{16} + 97962235 p^{5} T^{17} + 15643484 p^{6} T^{18} + 544940 p^{7} T^{19} + 97769 p^{8} T^{20} + 1781 p^{9} T^{21} + 430 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 13 T + 642 T^{2} + 7931 T^{3} + 208102 T^{4} + 2350248 T^{5} + 43881217 T^{6} + 448689818 T^{7} + 6656264326 T^{8} + 61306315608 T^{9} + 764644126911 T^{10} + 6302701751558 T^{11} + 68251618197003 T^{12} + 6302701751558 p T^{13} + 764644126911 p^{2} T^{14} + 61306315608 p^{3} T^{15} + 6656264326 p^{4} T^{16} + 448689818 p^{5} T^{17} + 43881217 p^{6} T^{18} + 2350248 p^{7} T^{19} + 208102 p^{8} T^{20} + 7931 p^{9} T^{21} + 642 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 6 T + 499 T^{2} + 1996 T^{3} + 124008 T^{4} + 380973 T^{5} + 21623828 T^{6} + 57244490 T^{7} + 2928997939 T^{8} + 6884893093 T^{9} + 3868084546 p T^{10} + 680931897969 T^{11} + 29175527008207 T^{12} + 680931897969 p T^{13} + 3868084546 p^{3} T^{14} + 6884893093 p^{3} T^{15} + 2928997939 p^{4} T^{16} + 57244490 p^{5} T^{17} + 21623828 p^{6} T^{18} + 380973 p^{7} T^{19} + 124008 p^{8} T^{20} + 1996 p^{9} T^{21} + 499 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 18 T + 746 T^{2} + 10931 T^{3} + 258147 T^{4} + 3211901 T^{5} + 56563606 T^{6} + 617030340 T^{7} + 8993867452 T^{8} + 87725667134 T^{9} + 1111985478215 T^{10} + 9767955242077 T^{11} + 110148098602543 T^{12} + 9767955242077 p T^{13} + 1111985478215 p^{2} T^{14} + 87725667134 p^{3} T^{15} + 8993867452 p^{4} T^{16} + 617030340 p^{5} T^{17} + 56563606 p^{6} T^{18} + 3211901 p^{7} T^{19} + 258147 p^{8} T^{20} + 10931 p^{9} T^{21} + 746 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 19 T + 267 T^{2} - 559 T^{3} - 14104 T^{4} + 276882 T^{5} + 496472 T^{6} - 28917735 T^{7} + 500203798 T^{8} - 2112028751 T^{9} + 11279268340 T^{10} + 94022750560 T^{11} + 142599766015 T^{12} + 94022750560 p T^{13} + 11279268340 p^{2} T^{14} - 2112028751 p^{3} T^{15} + 500203798 p^{4} T^{16} - 28917735 p^{5} T^{17} + 496472 p^{6} T^{18} + 276882 p^{7} T^{19} - 14104 p^{8} T^{20} - 559 p^{9} T^{21} + 267 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.84090057679338410976977057251, −2.69401571871987005747759398632, −2.68521501317936984400559923591, −2.60385686623885747115746653859, −2.49254830169972366917743053226, −2.39651140053058002641205596537, −2.35623822558099187481601478442, −2.31468441003083583489787242320, −2.24967063987078783656738625418, −2.23169925308801632632545809180, −2.20205646549572318890568156420, −2.20026704271647928990067821011, −2.19591024694940538466387669516, −1.73083448671008015376049695793, −1.62673597670332504914616970440, −1.62295742962564936711031312548, −1.61766363383493931085593227513, −1.61365560709753683396784329941, −1.52906997606395087447623975298, −1.46080527564898090919549969130, −1.44140051336084528774294600209, −1.36691452832794298290695684069, −1.35213795137843744330786339217, −1.18935626398359968054675704838, −1.16810420123449159401417922722, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.16810420123449159401417922722, 1.18935626398359968054675704838, 1.35213795137843744330786339217, 1.36691452832794298290695684069, 1.44140051336084528774294600209, 1.46080527564898090919549969130, 1.52906997606395087447623975298, 1.61365560709753683396784329941, 1.61766363383493931085593227513, 1.62295742962564936711031312548, 1.62673597670332504914616970440, 1.73083448671008015376049695793, 2.19591024694940538466387669516, 2.20026704271647928990067821011, 2.20205646549572318890568156420, 2.23169925308801632632545809180, 2.24967063987078783656738625418, 2.31468441003083583489787242320, 2.35623822558099187481601478442, 2.39651140053058002641205596537, 2.49254830169972366917743053226, 2.60385686623885747115746653859, 2.68521501317936984400559923591, 2.69401571871987005747759398632, 2.84090057679338410976977057251
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.