Dirichlet series
| L(s) = 1 | + 3.08e3·3-s − 1.48e8·7-s − 1.20e9·9-s − 2.33e9·11-s − 5.74e10·13-s + 4.65e11·17-s − 1.62e12·19-s − 4.56e11·21-s + 1.01e13·23-s − 2.38e13·27-s − 2.18e13·29-s − 6.95e13·31-s − 7.19e12·33-s − 2.15e15·37-s − 1.76e14·39-s + 1.36e15·41-s − 2.89e15·43-s − 1.74e15·47-s + 1.49e16·49-s + 1.43e15·51-s − 8.42e16·53-s − 5.00e15·57-s + 1.52e17·59-s + 2.11e17·61-s + 1.78e17·63-s − 6.56e17·67-s + 3.11e16·69-s + ⋯ |
| L(s) = 1 | + 0.0903·3-s − 1.38·7-s − 1.03·9-s − 0.298·11-s − 1.50·13-s + 0.952·17-s − 1.15·19-s − 0.125·21-s + 1.16·23-s − 0.602·27-s − 0.279·29-s − 0.472·31-s − 0.0269·33-s − 2.72·37-s − 0.135·39-s + 0.652·41-s − 0.878·43-s − 0.226·47-s + 1.31·49-s + 0.0860·51-s − 3.50·53-s − 0.104·57-s + 2.29·59-s + 2.31·61-s + 1.43·63-s − 2.94·67-s + 0.105·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(20-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{8} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.74126\times 10^{9}\) |
| Root analytic conductor: | \(15.1266\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 19/2, 19/2, 19/2, 19/2 ),\ 1 )\) |
Particular Values
| \(L(10)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{21}{2})\) | 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 | \( 1 \) | |
| 5 | \( 1 \) | ||
| good | 3 | $C_2 \wr S_4$ | \( 1 - 3080 T + 404444692 p T^{2} + 202914750200 p^{4} T^{3} + 991156259206306 p^{7} T^{4} + 202914750200 p^{23} T^{5} + 404444692 p^{39} T^{6} - 3080 p^{57} T^{7} + p^{76} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 148222040 T + 7015190281280620 T^{2} + \)\(23\!\cdots\!60\)\( p T^{3} + \)\(96\!\cdots\!86\)\( p^{3} T^{4} + \)\(23\!\cdots\!60\)\( p^{20} T^{5} + 7015190281280620 p^{38} T^{6} + 148222040 p^{57} T^{7} + p^{76} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 2334973920 T + 9101868096138635524 p T^{2} + \)\(11\!\cdots\!60\)\( p^{2} T^{3} + \)\(64\!\cdots\!06\)\( p^{3} T^{4} + \)\(11\!\cdots\!60\)\( p^{21} T^{5} + 9101868096138635524 p^{39} T^{6} + 2334973920 p^{57} T^{7} + p^{76} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 57423224120 T + \)\(26\!\cdots\!12\)\( p T^{2} + \)\(64\!\cdots\!00\)\( p^{2} T^{3} + \)\(25\!\cdots\!86\)\( p^{3} T^{4} + \)\(64\!\cdots\!00\)\( p^{21} T^{5} + \)\(26\!\cdots\!12\)\( p^{39} T^{6} + 57423224120 p^{57} T^{7} + p^{76} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 1612324440 p^{2} T + \)\(20\!\cdots\!96\)\( p^{2} T^{2} - \)\(70\!\cdots\!00\)\( p^{3} T^{3} + \)\(20\!\cdots\!62\)\( p^{4} T^{4} - \)\(70\!\cdots\!00\)\( p^{22} T^{5} + \)\(20\!\cdots\!96\)\( p^{40} T^{6} - 1612324440 p^{59} T^{7} + p^{76} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 85516745296 p T + \)\(51\!\cdots\!32\)\( T^{2} + \)\(48\!\cdots\!08\)\( p T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(48\!\cdots\!08\)\( p^{20} T^{5} + \)\(51\!\cdots\!32\)\( p^{38} T^{6} + 85516745296 p^{58} T^{7} + p^{76} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 439203029160 p T + \)\(35\!\cdots\!40\)\( p T^{2} - \)\(29\!\cdots\!60\)\( T^{3} + \)\(55\!\cdots\!38\)\( T^{4} - \)\(29\!\cdots\!60\)\( p^{19} T^{5} + \)\(35\!\cdots\!40\)\( p^{39} T^{6} - 439203029160 p^{58} T^{7} + p^{76} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 21802908180264 T + \)\(18\!\cdots\!12\)\( T^{2} + \)\(99\!\cdots\!08\)\( p T^{3} + \)\(14\!\cdots\!70\)\( T^{4} + \)\(99\!\cdots\!08\)\( p^{20} T^{5} + \)\(18\!\cdots\!12\)\( p^{38} T^{6} + 21802908180264 p^{57} T^{7} + p^{76} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 69517593805936 T + \)\(72\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!94\)\( T^{4} + \)\(39\!\cdots\!84\)\( p^{19} T^{5} + \)\(72\!\cdots\!20\)\( p^{38} T^{6} + 69517593805936 p^{57} T^{7} + p^{76} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 2156700568331960 T + \)\(35\!\cdots\!20\)\( T^{2} + \)\(38\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!58\)\( T^{4} + \)\(38\!\cdots\!80\)\( p^{19} T^{5} + \)\(35\!\cdots\!20\)\( p^{38} T^{6} + 2156700568331960 p^{57} T^{7} + p^{76} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 1367657291239944 T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(23\!\cdots\!76\)\( T^{3} + \)\(56\!\cdots\!14\)\( T^{4} - \)\(23\!\cdots\!76\)\( p^{19} T^{5} + \)\(10\!\cdots\!20\)\( p^{38} T^{6} - 1367657291239944 p^{57} T^{7} + p^{76} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 2895469359165560 T + \)\(11\!\cdots\!28\)\( T^{2} - \)\(28\!\cdots\!40\)\( T^{3} - \)\(79\!\cdots\!06\)\( T^{4} - \)\(28\!\cdots\!40\)\( p^{19} T^{5} + \)\(11\!\cdots\!28\)\( p^{38} T^{6} + 2895469359165560 p^{57} T^{7} + p^{76} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 1740670005708600 T + \)\(10\!\cdots\!24\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(77\!\cdots\!22\)\( T^{4} - \)\(17\!\cdots\!00\)\( p^{19} T^{5} + \)\(10\!\cdots\!24\)\( p^{38} T^{6} + 1740670005708600 p^{57} T^{7} + p^{76} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 84253756978305240 T + \)\(39\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!78\)\( T^{4} + \)\(12\!\cdots\!80\)\( p^{19} T^{5} + \)\(39\!\cdots\!80\)\( p^{38} T^{6} + 84253756978305240 p^{57} T^{7} + p^{76} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 152838143056415952 T + \)\(20\!\cdots\!20\)\( T^{2} - \)\(27\!\cdots\!08\)\( p T^{3} + \)\(12\!\cdots\!54\)\( T^{4} - \)\(27\!\cdots\!08\)\( p^{20} T^{5} + \)\(20\!\cdots\!20\)\( p^{38} T^{6} - 152838143056415952 p^{57} T^{7} + p^{76} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 211425306591350168 T + \)\(32\!\cdots\!48\)\( T^{2} - \)\(28\!\cdots\!16\)\( T^{3} + \)\(27\!\cdots\!70\)\( T^{4} - \)\(28\!\cdots\!16\)\( p^{19} T^{5} + \)\(32\!\cdots\!48\)\( p^{38} T^{6} - 211425306591350168 p^{57} T^{7} + p^{76} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 656919981274544360 T + \)\(32\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!18\)\( T^{4} + \)\(10\!\cdots\!80\)\( p^{19} T^{5} + \)\(32\!\cdots\!00\)\( p^{38} T^{6} + 656919981274544360 p^{57} T^{7} + p^{76} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 883122322400344752 T + \)\(70\!\cdots\!88\)\( T^{2} - \)\(35\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!70\)\( T^{4} - \)\(35\!\cdots\!24\)\( p^{19} T^{5} + \)\(70\!\cdots\!88\)\( p^{38} T^{6} - 883122322400344752 p^{57} T^{7} + p^{76} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 325226727237880520 T + \)\(63\!\cdots\!76\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!82\)\( T^{4} + \)\(20\!\cdots\!00\)\( p^{19} T^{5} + \)\(63\!\cdots\!76\)\( p^{38} T^{6} + 325226727237880520 p^{57} T^{7} + p^{76} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 850256855296398112 T + \)\(27\!\cdots\!80\)\( T^{2} + \)\(19\!\cdots\!92\)\( T^{3} + \)\(45\!\cdots\!74\)\( T^{4} + \)\(19\!\cdots\!92\)\( p^{19} T^{5} + \)\(27\!\cdots\!80\)\( p^{38} T^{6} + 850256855296398112 p^{57} T^{7} + p^{76} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 3602925283549230600 T + \)\(11\!\cdots\!40\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!18\)\( T^{4} - \)\(27\!\cdots\!00\)\( p^{19} T^{5} + \)\(11\!\cdots\!40\)\( p^{38} T^{6} - 3602925283549230600 p^{57} T^{7} + p^{76} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 418548299964421224 T + \)\(31\!\cdots\!52\)\( T^{2} + \)\(92\!\cdots\!88\)\( T^{3} + \)\(43\!\cdots\!70\)\( T^{4} + \)\(92\!\cdots\!88\)\( p^{19} T^{5} + \)\(31\!\cdots\!52\)\( p^{38} T^{6} - 418548299964421224 p^{57} T^{7} + p^{76} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 12135517460163032920 T + \)\(87\!\cdots\!00\)\( T^{2} - \)\(90\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(90\!\cdots\!60\)\( p^{19} T^{5} + \)\(87\!\cdots\!00\)\( p^{38} T^{6} - 12135517460163032920 p^{57} T^{7} + p^{76} T^{8} \) | |
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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
−7.26498506735434595177521344433, −7.17254098985511507531751144467, −6.75928089132399328868047346148, −6.65279330588605682237723941325, −6.56028963701900499190054120948, −6.01420242275800968941754781810, −5.71633959922927193406061047409, −5.51082545987165508203325878235, −5.35590378635479711096004511686, −5.03223400557548380806894255592, −4.64942793715293993615877041486, −4.62238758106430309496837153594, −4.07661927204939880083089120537, −3.60482473011096530847251139089, −3.49462007342634175953926490474, −3.33744192030966218176222550164, −3.18367934280541356551225596398, −2.63194143520683677645423081667, −2.39015485165275963389254412975, −2.30336473839566747926136666419, −2.11210808334441018040026468637, −1.48395069802586557660169314508, −1.25509236742631055205478686627, −1.11968145941346410751657196903, −0.59647529226507197354557514464, 0, 0, 0, 0, 0.59647529226507197354557514464, 1.11968145941346410751657196903, 1.25509236742631055205478686627, 1.48395069802586557660169314508, 2.11210808334441018040026468637, 2.30336473839566747926136666419, 2.39015485165275963389254412975, 2.63194143520683677645423081667, 3.18367934280541356551225596398, 3.33744192030966218176222550164, 3.49462007342634175953926490474, 3.60482473011096530847251139089, 4.07661927204939880083089120537, 4.62238758106430309496837153594, 4.64942793715293993615877041486, 5.03223400557548380806894255592, 5.35590378635479711096004511686, 5.51082545987165508203325878235, 5.71633959922927193406061047409, 6.01420242275800968941754781810, 6.56028963701900499190054120948, 6.65279330588605682237723941325, 6.75928089132399328868047346148, 7.17254098985511507531751144467, 7.26498506735434595177521344433