Dirichlet series
| L(s) = 1 | − 2·2-s + 5·4-s − 7·5-s − 7·7-s − 4·8-s − 9-s + 14·10-s − 8·11-s + 7·13-s + 14·14-s + 11·16-s + 2·18-s − 14·19-s − 35·20-s + 16·22-s − 2·23-s + 31·25-s − 14·26-s + 7·27-s − 35·28-s − 11·29-s − 14·31-s − 14·32-s + 49·35-s − 5·36-s − 30·37-s + 28·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 5/2·4-s − 3.13·5-s − 2.64·7-s − 1.41·8-s − 1/3·9-s + 4.42·10-s − 2.41·11-s + 1.94·13-s + 3.74·14-s + 11/4·16-s + 0.471·18-s − 3.21·19-s − 7.82·20-s + 3.41·22-s − 0.417·23-s + 31/5·25-s − 2.74·26-s + 1.34·27-s − 6.61·28-s − 2.04·29-s − 2.51·31-s − 2.47·32-s + 8.28·35-s − 5/6·36-s − 4.93·37-s + 4.54·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.28729\times 10^{-5}\) |
| Root analytic conductor: | \(0.625513\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.04052000036\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04052000036\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 + p T + 3 p T^{2} + 5 p T^{3} + 2 p^{2} T^{4} + 12 p^{2} T^{5} + 43 p^{2} T^{6} + 12 p^{3} T^{7} + 2 p^{4} T^{8} + 5 p^{4} T^{9} + 3 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| good | 2 | \( 1 + p T - T^{2} - p^{3} T^{3} - 7 p T^{4} + 35 T^{6} + 11 p^{2} T^{7} + p^{2} T^{8} - 9 p^{3} T^{9} - 43 p T^{10} + 35 T^{11} + 147 T^{12} + 35 p T^{13} - 43 p^{3} T^{14} - 9 p^{6} T^{15} + p^{6} T^{16} + 11 p^{7} T^{17} + 35 p^{6} T^{18} - 7 p^{9} T^{20} - p^{12} T^{21} - p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + T^{2} - 7 T^{3} + 2 p T^{4} - 7 T^{5} + 67 T^{6} - 14 T^{7} + p^{3} T^{8} - 49 p^{2} T^{9} + 32 p T^{10} - 245 T^{11} + 2107 T^{12} - 245 p T^{13} + 32 p^{3} T^{14} - 49 p^{5} T^{15} + p^{7} T^{16} - 14 p^{5} T^{17} + 67 p^{6} T^{18} - 7 p^{7} T^{19} + 2 p^{9} T^{20} - 7 p^{9} T^{21} + p^{10} T^{22} + p^{12} T^{24} \) | |
| 5 | \( 1 + 7 T + 18 T^{2} - 14 T^{3} - 254 T^{4} - 854 T^{5} - 976 T^{6} + 3129 T^{7} + 3488 p T^{8} + 36911 T^{9} + 12729 T^{10} - 168609 T^{11} - 575211 T^{12} - 168609 p T^{13} + 12729 p^{2} T^{14} + 36911 p^{3} T^{15} + 3488 p^{5} T^{16} + 3129 p^{5} T^{17} - 976 p^{6} T^{18} - 854 p^{7} T^{19} - 254 p^{8} T^{20} - 14 p^{9} T^{21} + 18 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 8 T + 5 T^{2} - 197 T^{3} - 931 T^{4} + 35 T^{5} + 16625 T^{6} + 60537 T^{7} - 46528 T^{8} - 987086 T^{9} - 2323147 T^{10} + 5151853 T^{11} + 40848171 T^{12} + 5151853 p T^{13} - 2323147 p^{2} T^{14} - 987086 p^{3} T^{15} - 46528 p^{4} T^{16} + 60537 p^{5} T^{17} + 16625 p^{6} T^{18} + 35 p^{7} T^{19} - 931 p^{8} T^{20} - 197 p^{9} T^{21} + 5 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 7 T + 2 T^{2} + 105 T^{3} - 74 T^{4} - 1484 T^{5} + 2174 T^{6} + 6181 T^{7} + 17379 T^{8} + 140 p^{2} T^{9} - 927256 T^{10} - 444087 T^{11} + 17334093 T^{12} - 444087 p T^{13} - 927256 p^{2} T^{14} + 140 p^{5} T^{15} + 17379 p^{4} T^{16} + 6181 p^{5} T^{17} + 2174 p^{6} T^{18} - 1484 p^{7} T^{19} - 74 p^{8} T^{20} + 105 p^{9} T^{21} + 2 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + T^{2} + 21 T^{3} - 351 T^{4} + 805 T^{5} + 1285 T^{6} + 21259 T^{7} + 15840 T^{8} + 89768 T^{9} + 494247 T^{10} - 5620153 T^{11} + 32910115 T^{12} - 5620153 p T^{13} + 494247 p^{2} T^{14} + 89768 p^{3} T^{15} + 15840 p^{4} T^{16} + 21259 p^{5} T^{17} + 1285 p^{6} T^{18} + 805 p^{7} T^{19} - 351 p^{8} T^{20} + 21 p^{9} T^{21} + p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( ( 1 + 7 T + 93 T^{2} + 455 T^{3} + 3770 T^{4} + 14868 T^{5} + 91939 T^{6} + 14868 p T^{7} + 3770 p^{2} T^{8} + 455 p^{3} T^{9} + 93 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 + 2 T - 36 T^{2} - 11 p T^{3} + 1043 T^{4} + 5789 T^{5} - 8190 T^{6} - 151793 T^{7} + 356927 T^{8} + 570344 T^{9} - 4524879 T^{10} - 17763963 T^{11} + 335310423 T^{12} - 17763963 p T^{13} - 4524879 p^{2} T^{14} + 570344 p^{3} T^{15} + 356927 p^{4} T^{16} - 151793 p^{5} T^{17} - 8190 p^{6} T^{18} + 5789 p^{7} T^{19} + 1043 p^{8} T^{20} - 11 p^{10} T^{21} - 36 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 11 T + 17 T^{2} - 449 T^{3} - 1393 T^{4} + 22610 T^{5} + 151697 T^{6} - 280710 T^{7} - 4342483 T^{8} + 10376917 T^{9} + 187164155 T^{10} + 92982673 T^{11} - 3869306847 T^{12} + 92982673 p T^{13} + 187164155 p^{2} T^{14} + 10376917 p^{3} T^{15} - 4342483 p^{4} T^{16} - 280710 p^{5} T^{17} + 151697 p^{6} T^{18} + 22610 p^{7} T^{19} - 1393 p^{8} T^{20} - 449 p^{9} T^{21} + 17 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 7 T + 88 T^{2} + 511 T^{3} + 4076 T^{4} + 21679 T^{5} + 134975 T^{6} + 21679 p T^{7} + 4076 p^{2} T^{8} + 511 p^{3} T^{9} + 88 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 + 30 T + 419 T^{2} + 3436 T^{3} + 15204 T^{4} - 19572 T^{5} - 958447 T^{6} - 8542504 T^{7} - 41955147 T^{8} - 80200122 T^{9} + 633349064 T^{10} + 8406169968 T^{11} + 60478612229 T^{12} + 8406169968 p T^{13} + 633349064 p^{2} T^{14} - 80200122 p^{3} T^{15} - 41955147 p^{4} T^{16} - 8542504 p^{5} T^{17} - 958447 p^{6} T^{18} - 19572 p^{7} T^{19} + 15204 p^{8} T^{20} + 3436 p^{9} T^{21} + 419 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 21 T + 114 T^{2} + 1015 T^{3} - 13416 T^{4} - 24332 T^{5} + 1124617 T^{6} - 5080138 T^{7} - 22183975 T^{8} + 257369588 T^{9} - 213592702 T^{10} - 7739726694 T^{11} + 66231437167 T^{12} - 7739726694 p T^{13} - 213592702 p^{2} T^{14} + 257369588 p^{3} T^{15} - 22183975 p^{4} T^{16} - 5080138 p^{5} T^{17} + 1124617 p^{6} T^{18} - 24332 p^{7} T^{19} - 13416 p^{8} T^{20} + 1015 p^{9} T^{21} + 114 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 17 T - 4 T^{2} + 1665 T^{3} - 8414 T^{4} - 31717 T^{5} + 362285 T^{6} - 1104421 T^{7} + 2730975 T^{8} - 12051566 T^{9} + 39133202 T^{10} + 1793387022 T^{11} - 23518305111 T^{12} + 1793387022 p T^{13} + 39133202 p^{2} T^{14} - 12051566 p^{3} T^{15} + 2730975 p^{4} T^{16} - 1104421 p^{5} T^{17} + 362285 p^{6} T^{18} - 31717 p^{7} T^{19} - 8414 p^{8} T^{20} + 1665 p^{9} T^{21} - 4 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 21 T + 256 T^{2} + 2793 T^{3} + 30210 T^{4} + 321587 T^{5} + 63677 p T^{6} + 24855299 T^{7} + 206144235 T^{8} + 1692381838 T^{9} + 13105466958 T^{10} + 93235046254 T^{11} + 634794782293 T^{12} + 93235046254 p T^{13} + 13105466958 p^{2} T^{14} + 1692381838 p^{3} T^{15} + 206144235 p^{4} T^{16} + 24855299 p^{5} T^{17} + 63677 p^{7} T^{18} + 321587 p^{7} T^{19} + 30210 p^{8} T^{20} + 2793 p^{9} T^{21} + 256 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 6 T - 142 T^{2} + 1098 T^{3} + 9891 T^{4} - 144396 T^{5} - 120260 T^{6} + 11267082 T^{7} - 36894535 T^{8} - 561327846 T^{9} + 4842655950 T^{10} + 11314891224 T^{11} - 319975006505 T^{12} + 11314891224 p T^{13} + 4842655950 p^{2} T^{14} - 561327846 p^{3} T^{15} - 36894535 p^{4} T^{16} + 11267082 p^{5} T^{17} - 120260 p^{6} T^{18} - 144396 p^{7} T^{19} + 9891 p^{8} T^{20} + 1098 p^{9} T^{21} - 142 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 14 T - 6 T^{2} + 49 T^{3} + 9743 T^{4} - 18893 T^{5} - 282222 T^{6} - 2654281 T^{7} - 4440451 T^{8} + 319262776 T^{9} + 353736969 T^{10} + 2345454825 T^{11} - 184991898189 T^{12} + 2345454825 p T^{13} + 353736969 p^{2} T^{14} + 319262776 p^{3} T^{15} - 4440451 p^{4} T^{16} - 2654281 p^{5} T^{17} - 282222 p^{6} T^{18} - 18893 p^{7} T^{19} + 9743 p^{8} T^{20} + 49 p^{9} T^{21} - 6 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 7 T + 18 T^{2} + 399 T^{3} + 5241 T^{4} + 65499 T^{5} + 509772 T^{6} + 1220030 T^{7} + 17894495 T^{8} + 360127180 T^{9} + 2886283711 T^{10} + 14727499143 T^{11} + 68550746229 T^{12} + 14727499143 p T^{13} + 2886283711 p^{2} T^{14} + 360127180 p^{3} T^{15} + 17894495 p^{4} T^{16} + 1220030 p^{5} T^{17} + 509772 p^{6} T^{18} + 65499 p^{7} T^{19} + 5241 p^{8} T^{20} + 399 p^{9} T^{21} + 18 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 24 T + 544 T^{2} - 7969 T^{3} + 103505 T^{4} - 1061603 T^{5} + 9586871 T^{6} - 1061603 p T^{7} + 103505 p^{2} T^{8} - 7969 p^{3} T^{9} + 544 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 + 39 T + 871 T^{2} + 14307 T^{3} + 190092 T^{4} + 2083466 T^{5} + 18915358 T^{6} + 138847559 T^{7} + 724402569 T^{8} + 765520348 T^{9} - 40173731301 T^{10} - 631486451309 T^{11} - 6236115389021 T^{12} - 631486451309 p T^{13} - 40173731301 p^{2} T^{14} + 765520348 p^{3} T^{15} + 724402569 p^{4} T^{16} + 138847559 p^{5} T^{17} + 18915358 p^{6} T^{18} + 2083466 p^{7} T^{19} + 190092 p^{8} T^{20} + 14307 p^{9} T^{21} + 871 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 42 T + 715 T^{2} - 5880 T^{3} + 11304 T^{4} + 423150 T^{5} - 8979169 T^{6} + 106207794 T^{7} - 735951847 T^{8} + 1892847096 T^{9} + 28699443498 T^{10} - 614746868148 T^{11} + 6731348891281 T^{12} - 614746868148 p T^{13} + 28699443498 p^{2} T^{14} + 1892847096 p^{3} T^{15} - 735951847 p^{4} T^{16} + 106207794 p^{5} T^{17} - 8979169 p^{6} T^{18} + 423150 p^{7} T^{19} + 11304 p^{8} T^{20} - 5880 p^{9} T^{21} + 715 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 8 T + 377 T^{2} + 2713 T^{3} + 66443 T^{4} + 395274 T^{5} + 6756729 T^{6} + 395274 p T^{7} + 66443 p^{2} T^{8} + 2713 p^{3} T^{9} + 377 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 7 T - 180 T^{2} - 2240 T^{3} + 4210 T^{4} + 197155 T^{5} + 2020379 T^{6} + 9004401 T^{7} - 183343253 T^{8} - 3038332654 T^{9} - 7239918141 T^{10} + 149580689517 T^{11} + 1902505655127 T^{12} + 149580689517 p T^{13} - 7239918141 p^{2} T^{14} - 3038332654 p^{3} T^{15} - 183343253 p^{4} T^{16} + 9004401 p^{5} T^{17} + 2020379 p^{6} T^{18} + 197155 p^{7} T^{19} + 4210 p^{8} T^{20} - 2240 p^{9} T^{21} - 180 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 14 T - 108 T^{2} - 343 T^{3} + 21712 T^{4} - 241381 T^{5} - 2769868 T^{6} + 37209816 T^{7} - 31055819 T^{8} - 3858106259 T^{9} + 34438376829 T^{10} + 162131963175 T^{11} - 4244781250857 T^{12} + 162131963175 p T^{13} + 34438376829 p^{2} T^{14} - 3858106259 p^{3} T^{15} - 31055819 p^{4} T^{16} + 37209816 p^{5} T^{17} - 2769868 p^{6} T^{18} - 241381 p^{7} T^{19} + 21712 p^{8} T^{20} - 343 p^{9} T^{21} - 108 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( ( 1 + 14 T + 554 T^{2} + 5824 T^{3} + 128360 T^{4} + 1046052 T^{5} + 16320585 T^{6} + 1046052 p T^{7} + 128360 p^{2} T^{8} + 5824 p^{3} T^{9} + 554 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−6.33932985382529677256975917358, −5.94108266850998434144706019974, −5.83382199925990007906099656357, −5.67849383471930044347932682662, −5.57194209151323007405562081865, −5.50634755821515084368061776778, −5.46736523349514573335157282478, −5.22684745679568487389320782640, −5.02164354716780403878355856088, −4.83606670085661235434745952908, −4.70822286006372818388281574800, −4.12185262540218666767795285872, −4.07621215475614994251748440219, −4.04278073394328058525189748727, −3.98082935971795124675138778587, −3.74065653944371111693780131818, −3.67569552583005435813613514120, −3.62887832335233405188024450214, −3.04596108039067770554766221350, −3.02981683616714777670576781868, −2.84429498269606879443041349591, −2.84183454217046821643758793370, −2.17489233707457700605520771934, −2.08511633403112429313801660238, −1.70047552364222161280466687435, 1.70047552364222161280466687435, 2.08511633403112429313801660238, 2.17489233707457700605520771934, 2.84183454217046821643758793370, 2.84429498269606879443041349591, 3.02981683616714777670576781868, 3.04596108039067770554766221350, 3.62887832335233405188024450214, 3.67569552583005435813613514120, 3.74065653944371111693780131818, 3.98082935971795124675138778587, 4.04278073394328058525189748727, 4.07621215475614994251748440219, 4.12185262540218666767795285872, 4.70822286006372818388281574800, 4.83606670085661235434745952908, 5.02164354716780403878355856088, 5.22684745679568487389320782640, 5.46736523349514573335157282478, 5.50634755821515084368061776778, 5.57194209151323007405562081865, 5.67849383471930044347932682662, 5.83382199925990007906099656357, 5.94108266850998434144706019974, 6.33932985382529677256975917358
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.