Dirichlet series
| L(s) = 1 | − 7.68e3·2-s − 2.01e7·4-s + 9.76e8·5-s + 2.49e10·7-s + 1.65e11·8-s − 7.50e12·10-s − 9.98e12·11-s − 1.43e14·13-s − 1.91e14·14-s + 1.23e15·16-s − 2.80e15·17-s + 1.50e16·19-s − 1.97e16·20-s + 7.67e16·22-s − 4.87e16·23-s + 5.96e17·25-s + 1.10e18·26-s − 5.02e17·28-s + 3.66e17·29-s + 5.26e18·31-s − 2.14e18·32-s + 2.15e19·34-s + 2.43e19·35-s − 8.25e19·37-s − 1.15e20·38-s + 1.61e20·40-s − 1.16e20·41-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 0.601·4-s + 1.78·5-s + 0.680·7-s + 0.850·8-s − 2.37·10-s − 0.959·11-s − 1.70·13-s − 0.902·14-s + 1.09·16-s − 1.16·17-s + 1.55·19-s − 1.07·20-s + 1.27·22-s − 0.464·23-s + 2·25-s + 2.26·26-s − 0.409·28-s + 0.192·29-s + 1.20·31-s − 0.328·32-s + 1.54·34-s + 1.21·35-s − 2.06·37-s − 2.06·38-s + 1.52·40-s − 0.804·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+25/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(4100625\) = \(3^{8} \cdot 5^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.00836\times 10^{9}\) |
| Root analytic conductor: | \(13.3491\) |
| Motivic weight: | \(25\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 4100625,\ (\ :25/2, 25/2, 25/2, 25/2),\ 1)\) |
Particular Values
| \(L(13)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{27}{2})\) | 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 | 3 | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - p^{12} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 7683 T + 19803275 p^{2} T^{2} + 9350685853 p^{6} T^{3} + 451253267883 p^{13} T^{4} + 9350685853 p^{31} T^{5} + 19803275 p^{52} T^{6} + 7683 p^{75} T^{7} + p^{100} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 508422464 p^{2} T + 53246328028610032828 p^{2} T^{2} - \)\(61\!\cdots\!88\)\( p^{5} T^{3} + \)\(94\!\cdots\!30\)\( p^{9} T^{4} - \)\(61\!\cdots\!88\)\( p^{30} T^{5} + 53246328028610032828 p^{52} T^{6} - 508422464 p^{77} T^{7} + p^{100} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 9989114586816 T + \)\(20\!\cdots\!12\)\( p T^{2} + \)\(67\!\cdots\!60\)\( p^{3} T^{3} + \)\(12\!\cdots\!86\)\( p^{5} T^{4} + \)\(67\!\cdots\!60\)\( p^{28} T^{5} + \)\(20\!\cdots\!12\)\( p^{51} T^{6} + 9989114586816 p^{75} T^{7} + p^{100} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 143366199676168 T + \)\(19\!\cdots\!96\)\( p^{2} T^{2} + \)\(17\!\cdots\!44\)\( p^{2} T^{3} + \)\(16\!\cdots\!10\)\( p^{3} T^{4} + \)\(17\!\cdots\!44\)\( p^{27} T^{5} + \)\(19\!\cdots\!96\)\( p^{52} T^{6} + 143366199676168 p^{75} T^{7} + p^{100} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 2801218397375112 T + \)\(11\!\cdots\!32\)\( T^{2} + \)\(16\!\cdots\!80\)\( p T^{3} + \)\(35\!\cdots\!54\)\( p^{2} T^{4} + \)\(16\!\cdots\!80\)\( p^{26} T^{5} + \)\(11\!\cdots\!32\)\( p^{50} T^{6} + 2801218397375112 p^{75} T^{7} + p^{100} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 15008254302340928 T + \)\(18\!\cdots\!08\)\( p T^{2} - \)\(96\!\cdots\!96\)\( p^{2} T^{3} + \)\(67\!\cdots\!66\)\( p^{3} T^{4} - \)\(96\!\cdots\!96\)\( p^{27} T^{5} + \)\(18\!\cdots\!08\)\( p^{51} T^{6} - 15008254302340928 p^{75} T^{7} + p^{100} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 48788561474090928 T + \)\(10\!\cdots\!36\)\( p T^{2} + \)\(26\!\cdots\!88\)\( p^{2} T^{3} + \)\(28\!\cdots\!30\)\( p^{3} T^{4} + \)\(26\!\cdots\!88\)\( p^{27} T^{5} + \)\(10\!\cdots\!36\)\( p^{51} T^{6} + 48788561474090928 p^{75} T^{7} + p^{100} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 366855400436538264 T + \)\(37\!\cdots\!60\)\( T^{2} - \)\(81\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!78\)\( T^{4} - \)\(81\!\cdots\!88\)\( p^{25} T^{5} + \)\(37\!\cdots\!60\)\( p^{50} T^{6} - 366855400436538264 p^{75} T^{7} + p^{100} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 5266475512689038864 T + \)\(25\!\cdots\!28\)\( T^{2} - \)\(36\!\cdots\!92\)\( T^{3} + \)\(33\!\cdots\!34\)\( p T^{4} - \)\(36\!\cdots\!92\)\( p^{25} T^{5} + \)\(25\!\cdots\!28\)\( p^{50} T^{6} - 5266475512689038864 p^{75} T^{7} + p^{100} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 82582977571412504584 T + \)\(49\!\cdots\!52\)\( T^{2} + \)\(21\!\cdots\!84\)\( T^{3} + \)\(90\!\cdots\!30\)\( T^{4} + \)\(21\!\cdots\!84\)\( p^{25} T^{5} + \)\(49\!\cdots\!52\)\( p^{50} T^{6} + 82582977571412504584 p^{75} T^{7} + p^{100} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(68\!\cdots\!68\)\( T^{2} + \)\(75\!\cdots\!92\)\( T^{3} + \)\(20\!\cdots\!34\)\( T^{4} + \)\(75\!\cdots\!92\)\( p^{25} T^{5} + \)\(68\!\cdots\!68\)\( p^{50} T^{6} + \)\(11\!\cdots\!44\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - \)\(29\!\cdots\!60\)\( T + \)\(24\!\cdots\!80\)\( T^{2} - \)\(55\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!98\)\( T^{4} - \)\(55\!\cdots\!80\)\( p^{25} T^{5} + \)\(24\!\cdots\!80\)\( p^{50} T^{6} - \)\(29\!\cdots\!60\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - \)\(25\!\cdots\!20\)\( T + \)\(11\!\cdots\!20\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + \)\(75\!\cdots\!98\)\( T^{4} + \)\(22\!\cdots\!60\)\( p^{25} T^{5} + \)\(11\!\cdots\!20\)\( p^{50} T^{6} - \)\(25\!\cdots\!20\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - \)\(17\!\cdots\!12\)\( T + \)\(37\!\cdots\!48\)\( T^{2} - \)\(50\!\cdots\!88\)\( T^{3} + \)\(65\!\cdots\!30\)\( T^{4} - \)\(50\!\cdots\!88\)\( p^{25} T^{5} + \)\(37\!\cdots\!48\)\( p^{50} T^{6} - \)\(17\!\cdots\!12\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + \)\(53\!\cdots\!32\)\( T + \)\(50\!\cdots\!92\)\( T^{2} + \)\(20\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!14\)\( T^{4} + \)\(20\!\cdots\!44\)\( p^{25} T^{5} + \)\(50\!\cdots\!92\)\( p^{50} T^{6} + \)\(53\!\cdots\!32\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(90\!\cdots\!76\)\( T^{2} - \)\(90\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!46\)\( T^{4} - \)\(90\!\cdots\!00\)\( p^{25} T^{5} + \)\(90\!\cdots\!76\)\( p^{50} T^{6} - \)\(19\!\cdots\!00\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - \)\(22\!\cdots\!72\)\( T + \)\(28\!\cdots\!72\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!40\)\( p^{25} T^{5} + \)\(28\!\cdots\!72\)\( p^{50} T^{6} - \)\(22\!\cdots\!72\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + \)\(82\!\cdots\!08\)\( T + \)\(59\!\cdots\!28\)\( T^{2} + \)\(37\!\cdots\!56\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(37\!\cdots\!56\)\( p^{25} T^{5} + \)\(59\!\cdots\!28\)\( p^{50} T^{6} + \)\(82\!\cdots\!08\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - \)\(55\!\cdots\!68\)\( T + \)\(19\!\cdots\!28\)\( T^{2} - \)\(45\!\cdots\!72\)\( T^{3} + \)\(94\!\cdots\!10\)\( T^{4} - \)\(45\!\cdots\!72\)\( p^{25} T^{5} + \)\(19\!\cdots\!28\)\( p^{50} T^{6} - \)\(55\!\cdots\!68\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - \)\(11\!\cdots\!60\)\( T + \)\(13\!\cdots\!96\)\( T^{2} - \)\(89\!\cdots\!20\)\( T^{3} + \)\(60\!\cdots\!06\)\( T^{4} - \)\(89\!\cdots\!20\)\( p^{25} T^{5} + \)\(13\!\cdots\!96\)\( p^{50} T^{6} - \)\(11\!\cdots\!60\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + \)\(24\!\cdots\!04\)\( T + \)\(51\!\cdots\!80\)\( T^{2} + \)\(68\!\cdots\!24\)\( T^{3} + \)\(79\!\cdots\!86\)\( T^{4} + \)\(68\!\cdots\!24\)\( p^{25} T^{5} + \)\(51\!\cdots\!80\)\( p^{50} T^{6} + \)\(24\!\cdots\!04\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + \)\(70\!\cdots\!08\)\( T + \)\(29\!\cdots\!92\)\( T^{2} + \)\(99\!\cdots\!96\)\( T^{3} + \)\(26\!\cdots\!14\)\( T^{4} + \)\(99\!\cdots\!96\)\( p^{25} T^{5} + \)\(29\!\cdots\!92\)\( p^{50} T^{6} + \)\(70\!\cdots\!08\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + \)\(15\!\cdots\!48\)\( T + \)\(25\!\cdots\!92\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!26\)\( T^{4} + \)\(22\!\cdots\!20\)\( p^{25} T^{5} + \)\(25\!\cdots\!92\)\( p^{50} T^{6} + \)\(15\!\cdots\!48\)\( p^{75} T^{7} + p^{100} T^{8} \) | |
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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
−8.261778601273865218635055797278, −7.55763149504390845720637833992, −7.40242982521219263200210055907, −7.00798891434045335509220859301, −6.78349234785790289689252463220, −6.50432857365175385161024787479, −5.98317206398814351064296627023, −5.93073559791205231896715917495, −5.36407160517188913990754188299, −5.10883254847517096989700727407, −5.03932731376465296767391536262, −4.89466557570648250987038425641, −4.70116037586858864969836496570, −3.88407381251661718001439524430, −3.73450594082398097344142526678, −3.61906667808783979400479432925, −2.84110472395185403962082277847, −2.66676425648311612699477196126, −2.47671670759645185353059708988, −2.24755251728973214491999018172, −2.07097706028519140559650085514, −1.46856057943459527341415855780, −1.10866205417874722541528976439, −1.10520762670988228496773997418, −0.894192834311552323981584345390, 0, 0, 0, 0, 0.894192834311552323981584345390, 1.10520762670988228496773997418, 1.10866205417874722541528976439, 1.46856057943459527341415855780, 2.07097706028519140559650085514, 2.24755251728973214491999018172, 2.47671670759645185353059708988, 2.66676425648311612699477196126, 2.84110472395185403962082277847, 3.61906667808783979400479432925, 3.73450594082398097344142526678, 3.88407381251661718001439524430, 4.70116037586858864969836496570, 4.89466557570648250987038425641, 5.03932731376465296767391536262, 5.10883254847517096989700727407, 5.36407160517188913990754188299, 5.93073559791205231896715917495, 5.98317206398814351064296627023, 6.50432857365175385161024787479, 6.78349234785790289689252463220, 7.00798891434045335509220859301, 7.40242982521219263200210055907, 7.55763149504390845720637833992, 8.261778601273865218635055797278