Dirichlet series
| L(s) = 1 | + 65·2-s + 860·3-s − 1.24e4·4-s + 5.59e4·6-s + 1.02e5·7-s − 9.65e5·8-s − 2.79e6·9-s + 6.67e6·11-s − 1.06e7·12-s + 4.92e6·13-s + 6.68e6·14-s + 6.13e7·16-s + 5.04e7·17-s − 1.81e8·18-s + 2.82e8·19-s + 8.84e7·21-s + 4.33e8·22-s + 1.20e9·23-s − 8.30e8·24-s + 3.20e8·26-s − 2.50e9·27-s − 1.27e9·28-s + 7.63e9·29-s − 1.27e10·31-s + 6.91e9·32-s + 5.74e9·33-s + 3.27e9·34-s + ⋯ |
| L(s) = 1 | + 0.718·2-s + 0.681·3-s − 1.51·4-s + 0.489·6-s + 0.330·7-s − 1.30·8-s − 1.75·9-s + 1.13·11-s − 1.03·12-s + 0.283·13-s + 0.237·14-s + 0.913·16-s + 0.506·17-s − 1.25·18-s + 1.37·19-s + 0.224·21-s + 0.815·22-s + 1.70·23-s − 0.886·24-s + 0.203·26-s − 1.24·27-s − 0.500·28-s + 2.38·29-s − 2.57·31-s + 1.13·32-s + 0.773·33-s + 0.363·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(390625\) = \(5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(516464.\) |
| Root analytic conductor: | \(5.17761\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 390625,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(6.528594962\) |
| \(L(\frac12)\) | \(\approx\) | \(6.528594962\) |
| \(L(\frac{15}{2})\) | 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 | 5 | \( 1 \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 65 T + 8315 p T^{2} - 115235 p^{3} T^{3} + 1110601 p^{7} T^{4} - 115235 p^{16} T^{5} + 8315 p^{27} T^{6} - 65 p^{39} T^{7} + p^{52} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - 860 T + 1178170 p T^{2} - 108970240 p^{3} T^{3} + 9061903127 p^{6} T^{4} - 108970240 p^{16} T^{5} + 1178170 p^{27} T^{6} - 860 p^{39} T^{7} + p^{52} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 - 102800 T + 140026314100 T^{2} - 1578231845602800 p T^{3} + \)\(31\!\cdots\!02\)\( p^{2} T^{4} - 1578231845602800 p^{14} T^{5} + 140026314100 p^{26} T^{6} - 102800 p^{39} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 6675428 T + 2756457151538 p T^{2} + 94821398286710144 p^{2} T^{3} + \)\(23\!\cdots\!45\)\( p^{3} T^{4} + 94821398286710144 p^{15} T^{5} + 2756457151538 p^{27} T^{6} - 6675428 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 4926920 T + 57063151297540 p T^{2} - \)\(46\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!18\)\( T^{4} - \)\(46\!\cdots\!60\)\( p^{13} T^{5} + 57063151297540 p^{27} T^{6} - 4926920 p^{39} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 2965660 p T + 18594247455561490 T^{2} + \)\(19\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!63\)\( T^{4} + \)\(19\!\cdots\!60\)\( p^{13} T^{5} + 18594247455561490 p^{26} T^{6} - 2965660 p^{40} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 282648860 T + 66086749832775886 T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!11\)\( T^{4} - \)\(13\!\cdots\!20\)\( p^{13} T^{5} + 66086749832775886 p^{26} T^{6} - 282648860 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 1208437080 T + 2202623318568962180 T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!78\)\( T^{4} - \)\(17\!\cdots\!40\)\( p^{13} T^{5} + 2202623318568962180 p^{26} T^{6} - 1208437080 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 7630648840 T + 44920523629086223156 T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(73\!\cdots\!26\)\( T^{4} - \)\(19\!\cdots\!80\)\( p^{13} T^{5} + 44920523629086223156 p^{26} T^{6} - 7630648840 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 12716039032 T + \)\(11\!\cdots\!48\)\( T^{2} + \)\(76\!\cdots\!84\)\( T^{3} + \)\(40\!\cdots\!70\)\( T^{4} + \)\(76\!\cdots\!84\)\( p^{13} T^{5} + \)\(11\!\cdots\!48\)\( p^{26} T^{6} + 12716039032 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 38137910960 T + \)\(12\!\cdots\!20\)\( T^{2} - \)\(25\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!14\)\( p T^{4} - \)\(25\!\cdots\!20\)\( p^{13} T^{5} + \)\(12\!\cdots\!20\)\( p^{26} T^{6} - 38137910960 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 81254933612 T + \)\(52\!\cdots\!38\)\( T^{2} + \)\(23\!\cdots\!64\)\( T^{3} + \)\(80\!\cdots\!95\)\( T^{4} + \)\(23\!\cdots\!64\)\( p^{13} T^{5} + \)\(52\!\cdots\!38\)\( p^{26} T^{6} + 81254933612 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 97224763400 T + \)\(56\!\cdots\!00\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(85\!\cdots\!98\)\( T^{4} - \)\(21\!\cdots\!00\)\( p^{13} T^{5} + \)\(56\!\cdots\!00\)\( p^{26} T^{6} - 97224763400 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 69940090280 T + \)\(15\!\cdots\!60\)\( T^{2} - \)\(60\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} - \)\(60\!\cdots\!60\)\( p^{13} T^{5} + \)\(15\!\cdots\!60\)\( p^{26} T^{6} - 69940090280 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 179518658440 T + \)\(87\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!58\)\( T^{4} + \)\(11\!\cdots\!20\)\( p^{13} T^{5} + \)\(87\!\cdots\!60\)\( p^{26} T^{6} + 179518658440 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 858015815320 T + \)\(50\!\cdots\!16\)\( T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(69\!\cdots\!46\)\( T^{4} + \)\(19\!\cdots\!40\)\( p^{13} T^{5} + \)\(50\!\cdots\!16\)\( p^{26} T^{6} + 858015815320 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 589205515328 T + \)\(45\!\cdots\!68\)\( T^{2} - \)\(20\!\cdots\!76\)\( T^{3} + \)\(99\!\cdots\!70\)\( T^{4} - \)\(20\!\cdots\!76\)\( p^{13} T^{5} + \)\(45\!\cdots\!68\)\( p^{26} T^{6} - 589205515328 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 685669480180 T + \)\(19\!\cdots\!90\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!63\)\( T^{4} + \)\(10\!\cdots\!60\)\( p^{13} T^{5} + \)\(19\!\cdots\!90\)\( p^{26} T^{6} + 685669480180 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 949682379448 T + \)\(18\!\cdots\!08\)\( T^{2} + \)\(36\!\cdots\!04\)\( T^{3} + \)\(72\!\cdots\!70\)\( T^{4} + \)\(36\!\cdots\!04\)\( p^{13} T^{5} + \)\(18\!\cdots\!08\)\( p^{26} T^{6} - 949682379448 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1688808513820 T + \)\(43\!\cdots\!30\)\( T^{2} + \)\(55\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!03\)\( T^{4} + \)\(55\!\cdots\!60\)\( p^{13} T^{5} + \)\(43\!\cdots\!30\)\( p^{26} T^{6} + 1688808513820 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 355815036160 T + \)\(10\!\cdots\!56\)\( T^{2} - \)\(51\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!26\)\( T^{4} - \)\(51\!\cdots\!80\)\( p^{13} T^{5} + \)\(10\!\cdots\!56\)\( p^{26} T^{6} + 355815036160 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 8420327909340 T + \)\(50\!\cdots\!90\)\( T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + \)\(69\!\cdots\!63\)\( T^{4} - \)\(20\!\cdots\!20\)\( p^{13} T^{5} + \)\(50\!\cdots\!90\)\( p^{26} T^{6} - 8420327909340 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 2506082642820 T + \)\(18\!\cdots\!26\)\( T^{2} - \)\(38\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!91\)\( T^{4} - \)\(38\!\cdots\!40\)\( p^{13} T^{5} + \)\(18\!\cdots\!26\)\( p^{26} T^{6} - 2506082642820 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 12121501720280 T + \)\(25\!\cdots\!60\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!58\)\( T^{4} - \)\(20\!\cdots\!60\)\( p^{13} T^{5} + \)\(25\!\cdots\!60\)\( p^{26} T^{6} - 12121501720280 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−9.951453111740709402759355080247, −9.481855414186041404280289658780, −9.157832653235535370461993912446, −9.095556060329120711700539663596, −8.926777477133739481312818020199, −8.256957303338856991702881208139, −8.233112681367478813318773100606, −7.53572344434179092622926340368, −7.47552573014934341149771438349, −6.58449641743793327434933752416, −6.38119594120128070563526322898, −5.73255632382132926645426323241, −5.60787205011901559281531384364, −5.06512132163117027679297248528, −4.66280611262397693392317454731, −4.53890859761253918227887178818, −3.94591360760522382672641387616, −3.20598038026399978015657660062, −3.19570678541605885675553922147, −3.10536180287621335628610470936, −2.24570101778188134420839958875, −1.64639031407517314017137256247, −1.06423796418252360496236849400, −0.65802632518984988939965817206, −0.45532495023198503889960565765, 0.45532495023198503889960565765, 0.65802632518984988939965817206, 1.06423796418252360496236849400, 1.64639031407517314017137256247, 2.24570101778188134420839958875, 3.10536180287621335628610470936, 3.19570678541605885675553922147, 3.20598038026399978015657660062, 3.94591360760522382672641387616, 4.53890859761253918227887178818, 4.66280611262397693392317454731, 5.06512132163117027679297248528, 5.60787205011901559281531384364, 5.73255632382132926645426323241, 6.38119594120128070563526322898, 6.58449641743793327434933752416, 7.47552573014934341149771438349, 7.53572344434179092622926340368, 8.233112681367478813318773100606, 8.256957303338856991702881208139, 8.926777477133739481312818020199, 9.095556060329120711700539663596, 9.157832653235535370461993912446, 9.481855414186041404280289658780, 9.951453111740709402759355080247