Dirichlet series
| L(s) = 1 | − 1.03e11·2-s + 2.36e15·3-s + 7.08e21·4-s + 4.75e24·5-s − 2.43e26·6-s − 6.85e29·7-s − 4.05e32·8-s − 1.35e34·9-s − 4.89e35·10-s + 3.96e36·11-s + 1.67e37·12-s + 2.52e39·13-s + 7.06e40·14-s + 1.12e40·15-s + 2.09e43·16-s + 3.54e43·17-s + 1.39e45·18-s + 2.84e45·19-s + 3.36e46·20-s − 1.62e45·21-s − 4.09e47·22-s + 3.72e47·23-s − 9.58e47·24-s − 1.72e49·25-s − 2.60e50·26-s − 3.53e50·27-s − 4.85e51·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.0272·3-s + 3·4-s + 0.730·5-s − 0.0578·6-s − 0.684·7-s − 3.53·8-s − 1.79·9-s − 1.54·10-s + 0.425·11-s + 0.0817·12-s + 0.721·13-s + 1.45·14-s + 0.0199·15-s + 15/4·16-s + 0.739·17-s + 3.81·18-s + 1.14·19-s + 2.19·20-s − 0.0186·21-s − 0.903·22-s + 0.169·23-s − 0.0963·24-s − 0.408·25-s − 1.52·26-s − 0.543·27-s − 2.05·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+71/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(260295.\) |
| Root analytic conductor: | \(7.99057\) |
| Motivic weight: | \(71\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 8,\ (\ :71/2, 71/2, 71/2),\ 1)\) |
Particular Values
| \(L(36)\) | \(\approx\) | \(1.693673841\) |
| \(L(\frac12)\) | \(\approx\) | \(1.693673841\) |
| \(L(\frac{73}{2})\) | 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 | $C_1$ | \( ( 1 + p^{35} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 262507810300804 p^{2} T + \)\(22\!\cdots\!77\)\( p^{10} T^{2} + \)\(34\!\cdots\!16\)\( p^{25} T^{3} + \)\(22\!\cdots\!77\)\( p^{81} T^{4} - 262507810300804 p^{144} T^{5} + p^{213} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - \)\(19\!\cdots\!66\)\( p^{2} T + \)\(51\!\cdots\!51\)\( p^{7} T^{2} - \)\(12\!\cdots\!84\)\( p^{15} T^{3} + \)\(51\!\cdots\!51\)\( p^{78} T^{4} - \)\(19\!\cdots\!66\)\( p^{144} T^{5} + p^{213} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!28\)\( p^{2} T + \)\(39\!\cdots\!57\)\( p^{8} T^{2} + \)\(40\!\cdots\!16\)\( p^{16} T^{3} + \)\(39\!\cdots\!57\)\( p^{79} T^{4} + \)\(13\!\cdots\!28\)\( p^{144} T^{5} + p^{213} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 - \)\(36\!\cdots\!96\)\( p T + \)\(62\!\cdots\!95\)\( p^{3} T^{2} - \)\(33\!\cdots\!40\)\( p^{8} T^{3} + \)\(62\!\cdots\!95\)\( p^{74} T^{4} - \)\(36\!\cdots\!96\)\( p^{143} T^{5} + p^{213} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(19\!\cdots\!42\)\( p T + \)\(47\!\cdots\!03\)\( p^{4} T^{2} - \)\(41\!\cdots\!64\)\( p^{9} T^{3} + \)\(47\!\cdots\!03\)\( p^{75} T^{4} - \)\(19\!\cdots\!42\)\( p^{143} T^{5} + p^{213} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(35\!\cdots\!18\)\( T + \)\(10\!\cdots\!63\)\( p^{2} T^{2} - \)\(54\!\cdots\!72\)\( p^{5} T^{3} + \)\(10\!\cdots\!63\)\( p^{73} T^{4} - \)\(35\!\cdots\!18\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 - \)\(28\!\cdots\!20\)\( T + \)\(52\!\cdots\!37\)\( p^{2} T^{2} - \)\(14\!\cdots\!40\)\( p^{5} T^{3} + \)\(52\!\cdots\!37\)\( p^{73} T^{4} - \)\(28\!\cdots\!20\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(16\!\cdots\!92\)\( p T + \)\(23\!\cdots\!13\)\( p^{4} T^{2} + \)\(61\!\cdots\!16\)\( p^{5} T^{3} + \)\(23\!\cdots\!13\)\( p^{75} T^{4} - \)\(16\!\cdots\!92\)\( p^{143} T^{5} + p^{213} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(28\!\cdots\!70\)\( T + \)\(10\!\cdots\!07\)\( p^{2} T^{2} + \)\(63\!\cdots\!60\)\( p^{3} T^{3} + \)\(10\!\cdots\!07\)\( p^{73} T^{4} - \)\(28\!\cdots\!70\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 + \)\(19\!\cdots\!44\)\( p T + \)\(77\!\cdots\!75\)\( p^{3} T^{2} + \)\(31\!\cdots\!40\)\( p^{5} T^{3} + \)\(77\!\cdots\!75\)\( p^{74} T^{4} + \)\(19\!\cdots\!44\)\( p^{143} T^{5} + p^{213} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(68\!\cdots\!62\)\( T + \)\(20\!\cdots\!51\)\( p T^{2} + \)\(22\!\cdots\!04\)\( p^{2} T^{3} + \)\(20\!\cdots\!51\)\( p^{72} T^{4} + \)\(68\!\cdots\!62\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(25\!\cdots\!86\)\( T + \)\(93\!\cdots\!55\)\( T^{2} - \)\(37\!\cdots\!80\)\( p T^{3} + \)\(93\!\cdots\!55\)\( p^{71} T^{4} - \)\(25\!\cdots\!86\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 - \)\(18\!\cdots\!36\)\( T + \)\(90\!\cdots\!71\)\( p T^{2} - \)\(20\!\cdots\!68\)\( p^{2} T^{3} + \)\(90\!\cdots\!71\)\( p^{72} T^{4} - \)\(18\!\cdots\!36\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(43\!\cdots\!32\)\( T + \)\(12\!\cdots\!11\)\( p T^{2} + \)\(45\!\cdots\!64\)\( p^{2} T^{3} + \)\(12\!\cdots\!11\)\( p^{72} T^{4} + \)\(43\!\cdots\!32\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(30\!\cdots\!06\)\( T + \)\(14\!\cdots\!51\)\( p T^{2} - \)\(53\!\cdots\!08\)\( p^{2} T^{3} + \)\(14\!\cdots\!51\)\( p^{72} T^{4} - \)\(30\!\cdots\!06\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(46\!\cdots\!40\)\( T + \)\(18\!\cdots\!03\)\( p T^{2} - \)\(37\!\cdots\!20\)\( p^{2} T^{3} + \)\(18\!\cdots\!03\)\( p^{72} T^{4} - \)\(46\!\cdots\!40\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 - \)\(27\!\cdots\!06\)\( p T + \)\(12\!\cdots\!35\)\( p^{2} T^{2} - \)\(58\!\cdots\!00\)\( p^{3} T^{3} + \)\(12\!\cdots\!35\)\( p^{73} T^{4} - \)\(27\!\cdots\!06\)\( p^{143} T^{5} + p^{213} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 - \)\(43\!\cdots\!24\)\( p T + \)\(92\!\cdots\!33\)\( p^{2} T^{2} - \)\(11\!\cdots\!68\)\( p^{3} T^{3} + \)\(92\!\cdots\!33\)\( p^{73} T^{4} - \)\(43\!\cdots\!24\)\( p^{143} T^{5} + p^{213} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 - \)\(91\!\cdots\!96\)\( T + \)\(80\!\cdots\!85\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!85\)\( p^{71} T^{4} - \)\(91\!\cdots\!96\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 - \)\(28\!\cdots\!86\)\( T + \)\(59\!\cdots\!63\)\( T^{2} - \)\(84\!\cdots\!72\)\( T^{3} + \)\(59\!\cdots\!63\)\( p^{71} T^{4} - \)\(28\!\cdots\!86\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(35\!\cdots\!40\)\( T + \)\(17\!\cdots\!37\)\( T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!37\)\( p^{71} T^{4} - \)\(35\!\cdots\!40\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 + \)\(49\!\cdots\!64\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(21\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!33\)\( p^{71} T^{4} + \)\(49\!\cdots\!64\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(17\!\cdots\!70\)\( T + \)\(72\!\cdots\!67\)\( T^{2} + \)\(90\!\cdots\!60\)\( T^{3} + \)\(72\!\cdots\!67\)\( p^{71} T^{4} + \)\(17\!\cdots\!70\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 - \)\(41\!\cdots\!58\)\( T + \)\(31\!\cdots\!47\)\( T^{2} - \)\(74\!\cdots\!04\)\( T^{3} + \)\(31\!\cdots\!47\)\( p^{71} T^{4} - \)\(41\!\cdots\!58\)\( p^{142} T^{5} + p^{213} T^{6} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.02150775375769971640294475246, −11.35044125004675883212595408967, −11.14109875602125567823340233419, −10.73768234587935097249065419450, −9.922385362819168541040472256629, −9.602407719289499622842537816045, −9.390456928277208564879103718363, −8.861465666356790097821213590195, −8.309308116085308896947760536579, −8.076428026555780078305276314092, −7.46402528121473953105970301410, −6.70007701866108697943991357577, −6.63912488618128702132546548529, −5.75985300194031720340061788654, −5.62805751643630772634374995411, −5.29138069349547725323626923255, −3.91079472184107988548123664514, −3.41922911598692545131697169932, −3.17556865280986370615013316643, −2.35575346106003753127284087981, −2.27551970500632059377150467691, −1.66064879833718373124201852285, −0.932287180389301422681322737017, −0.70844749435404216599064018316, −0.41980751051285742381739687448, 0.41980751051285742381739687448, 0.70844749435404216599064018316, 0.932287180389301422681322737017, 1.66064879833718373124201852285, 2.27551970500632059377150467691, 2.35575346106003753127284087981, 3.17556865280986370615013316643, 3.41922911598692545131697169932, 3.91079472184107988548123664514, 5.29138069349547725323626923255, 5.62805751643630772634374995411, 5.75985300194031720340061788654, 6.63912488618128702132546548529, 6.70007701866108697943991357577, 7.46402528121473953105970301410, 8.076428026555780078305276314092, 8.309308116085308896947760536579, 8.861465666356790097821213590195, 9.390456928277208564879103718363, 9.602407719289499622842537816045, 9.922385362819168541040472256629, 10.73768234587935097249065419450, 11.14109875602125567823340233419, 11.35044125004675883212595408967, 12.02150775375769971640294475246