Dirichlet series
| L(s) = 1 | − 512·2-s + 1.63e5·4-s − 2.91e5·5-s + 3.29e6·7-s − 4.19e7·8-s + 1.49e8·10-s − 1.21e8·11-s + 1.70e8·13-s − 1.68e9·14-s + 9.39e9·16-s − 3.45e8·17-s + 4.73e9·19-s − 4.77e10·20-s + 6.20e10·22-s − 1.70e10·23-s + 5.60e8·25-s − 8.71e10·26-s + 5.39e11·28-s − 9.62e10·29-s + 6.56e10·31-s − 1.92e12·32-s + 1.76e11·34-s − 9.59e11·35-s + 2.79e11·37-s − 2.42e12·38-s + 1.22e13·40-s − 2.74e11·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 1.66·5-s + 1.51·7-s − 7.07·8-s + 4.71·10-s − 1.87·11-s + 0.752·13-s − 4.27·14-s + 35/4·16-s − 0.204·17-s + 1.21·19-s − 8.33·20-s + 5.30·22-s − 1.04·23-s + 0.0183·25-s − 2.12·26-s + 7.55·28-s − 1.03·29-s + 0.428·31-s − 9.89·32-s + 0.577·34-s − 2.52·35-s + 0.484·37-s − 3.44·38-s + 11.7·40-s − 0.220·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{8} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.04495\times 10^{9}\) |
| Root analytic conductor: | \(13.4087\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 15/2, 15/2, 15/2, 15/2 ),\ 1 )\) |
Particular Values
| \(L(8)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{17}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) |
| 3 | \( 1 \) | ||
| 7 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 291312 T + 16860532132 p T^{2} + 686482401873744 p^{2} T^{3} + 27284728372873500126 p^{3} T^{4} + 686482401873744 p^{17} T^{5} + 16860532132 p^{31} T^{6} + 291312 p^{45} T^{7} + p^{60} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 121144752 T + 657292329014020 p T^{2} + \)\(49\!\cdots\!76\)\( p^{2} T^{3} + \)\(39\!\cdots\!98\)\( p^{3} T^{4} + \)\(49\!\cdots\!76\)\( p^{17} T^{5} + 657292329014020 p^{31} T^{6} + 121144752 p^{45} T^{7} + p^{60} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 170165240 T + 8613511758903868 p T^{2} - \)\(87\!\cdots\!16\)\( p^{2} T^{3} + \)\(35\!\cdots\!70\)\( p^{3} T^{4} - \)\(87\!\cdots\!16\)\( p^{17} T^{5} + 8613511758903868 p^{31} T^{6} - 170165240 p^{45} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 345685200 T + 4867623508292627396 T^{2} + \)\(88\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(88\!\cdots\!48\)\( p^{15} T^{5} + 4867623508292627396 p^{30} T^{6} + 345685200 p^{45} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 4738793696 T + 55770422627409479884 T^{2} - \)\(21\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} - \)\(21\!\cdots\!16\)\( p^{15} T^{5} + 55770422627409479884 p^{30} T^{6} - 4738793696 p^{45} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 17023827024 T + \)\(62\!\cdots\!08\)\( T^{2} + \)\(14\!\cdots\!36\)\( T^{3} + \)\(19\!\cdots\!14\)\( T^{4} + \)\(14\!\cdots\!36\)\( p^{15} T^{5} + \)\(62\!\cdots\!08\)\( p^{30} T^{6} + 17023827024 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 96242392128 T + \)\(60\!\cdots\!44\)\( T^{2} + \)\(99\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} + \)\(99\!\cdots\!44\)\( p^{15} T^{5} + \)\(60\!\cdots\!44\)\( p^{30} T^{6} + 96242392128 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 65665329632 T + \)\(67\!\cdots\!00\)\( T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(21\!\cdots\!46\)\( T^{4} - \)\(39\!\cdots\!44\)\( p^{15} T^{5} + \)\(67\!\cdots\!00\)\( p^{30} T^{6} - 65665329632 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 279650496248 T + \)\(84\!\cdots\!60\)\( T^{2} - \)\(14\!\cdots\!28\)\( T^{3} + \)\(36\!\cdots\!82\)\( T^{4} - \)\(14\!\cdots\!28\)\( p^{15} T^{5} + \)\(84\!\cdots\!60\)\( p^{30} T^{6} - 279650496248 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 274507805040 T + \)\(49\!\cdots\!32\)\( T^{2} + \)\(17\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!02\)\( T^{4} + \)\(17\!\cdots\!44\)\( p^{15} T^{5} + \)\(49\!\cdots\!32\)\( p^{30} T^{6} + 274507805040 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 1551932654480 T + \)\(96\!\cdots\!56\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!82\)\( T^{4} - \)\(11\!\cdots\!00\)\( p^{15} T^{5} + \)\(96\!\cdots\!56\)\( p^{30} T^{6} - 1551932654480 p^{45} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 686050223712 T + \)\(16\!\cdots\!92\)\( T^{2} - \)\(59\!\cdots\!88\)\( T^{3} + \)\(60\!\cdots\!94\)\( T^{4} - \)\(59\!\cdots\!88\)\( p^{15} T^{5} + \)\(16\!\cdots\!92\)\( p^{30} T^{6} + 686050223712 p^{45} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 4929897017760 T + \)\(24\!\cdots\!00\)\( T^{2} + \)\(94\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!98\)\( T^{4} + \)\(94\!\cdots\!20\)\( p^{15} T^{5} + \)\(24\!\cdots\!00\)\( p^{30} T^{6} + 4929897017760 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 15661604798112 T + \)\(14\!\cdots\!28\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(79\!\cdots\!46\)\( T^{4} + \)\(16\!\cdots\!40\)\( p^{15} T^{5} + \)\(14\!\cdots\!28\)\( p^{30} T^{6} + 15661604798112 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 7270740102248 T + \)\(16\!\cdots\!76\)\( T^{2} - \)\(14\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} - \)\(14\!\cdots\!12\)\( p^{15} T^{5} + \)\(16\!\cdots\!76\)\( p^{30} T^{6} - 7270740102248 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 2642113680560 T + \)\(95\!\cdots\!64\)\( T^{2} - \)\(19\!\cdots\!92\)\( T^{3} + \)\(35\!\cdots\!26\)\( T^{4} - \)\(19\!\cdots\!92\)\( p^{15} T^{5} + \)\(95\!\cdots\!64\)\( p^{30} T^{6} - 2642113680560 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 62559334721520 T + \)\(73\!\cdots\!20\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!38\)\( T^{4} + \)\(41\!\cdots\!00\)\( p^{15} T^{5} + \)\(73\!\cdots\!20\)\( p^{30} T^{6} + 62559334721520 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 53486860270312 T + \)\(81\!\cdots\!88\)\( T^{2} - \)\(21\!\cdots\!84\)\( T^{3} + \)\(71\!\cdots\!54\)\( T^{4} - \)\(21\!\cdots\!84\)\( p^{15} T^{5} + \)\(81\!\cdots\!88\)\( p^{30} T^{6} + 53486860270312 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 77560377412544 T + \)\(63\!\cdots\!04\)\( T^{2} - \)\(70\!\cdots\!44\)\( T^{3} + \)\(19\!\cdots\!46\)\( T^{4} - \)\(70\!\cdots\!44\)\( p^{15} T^{5} + \)\(63\!\cdots\!04\)\( p^{30} T^{6} - 77560377412544 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 58088393760768 T + \)\(18\!\cdots\!04\)\( T^{2} - \)\(16\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} - \)\(16\!\cdots\!08\)\( p^{15} T^{5} + \)\(18\!\cdots\!04\)\( p^{30} T^{6} - 58088393760768 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 296914762526928 T + \)\(71\!\cdots\!72\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(18\!\cdots\!22\)\( T^{4} - \)\(15\!\cdots\!08\)\( p^{15} T^{5} + \)\(71\!\cdots\!72\)\( p^{30} T^{6} - 296914762526928 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 1391306483388248 T + \)\(18\!\cdots\!72\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!18\)\( T^{4} - \)\(20\!\cdots\!44\)\( p^{15} T^{5} + \)\(18\!\cdots\!72\)\( p^{30} T^{6} - 1391306483388248 p^{45} T^{7} + p^{60} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.897259853249197629428821572269, −7.52156923808356174634982451929, −7.32072244245428704881919760248, −7.25879874903563123607624127663, −6.93683553238669766441544072755, −6.15266239233384686371191986818, −6.06785298738984309778946827642, −5.97429852918916902751384281765, −5.67225165670080194763837073374, −5.19173586574533876476882586029, −4.71480409628724996478624117168, −4.58174239987606674380146150064, −4.56649057068298715281244480268, −3.59376104113983306732956849581, −3.53441879706584360057718021423, −3.48292655177590817079583563173, −3.19639326905155308868233569554, −2.37780342062282565858188240732, −2.21650574198680803720491506879, −2.21542333557859164724582698828, −2.10802392172512468552736555383, −1.25109310323969284290957827778, −1.21072739991671890353147402632, −1.02451729792170880937997319829, −0.832302871713273042344632083406, 0, 0, 0, 0, 0.832302871713273042344632083406, 1.02451729792170880937997319829, 1.21072739991671890353147402632, 1.25109310323969284290957827778, 2.10802392172512468552736555383, 2.21542333557859164724582698828, 2.21650574198680803720491506879, 2.37780342062282565858188240732, 3.19639326905155308868233569554, 3.48292655177590817079583563173, 3.53441879706584360057718021423, 3.59376104113983306732956849581, 4.56649057068298715281244480268, 4.58174239987606674380146150064, 4.71480409628724996478624117168, 5.19173586574533876476882586029, 5.67225165670080194763837073374, 5.97429852918916902751384281765, 6.06785298738984309778946827642, 6.15266239233384686371191986818, 6.93683553238669766441544072755, 7.25879874903563123607624127663, 7.32072244245428704881919760248, 7.52156923808356174634982451929, 7.897259853249197629428821572269