Dirichlet series
| L(s) = 1 | − 2.91e3·2-s − 8.32e4·3-s + 4.62e6·4-s + 2.42e8·6-s − 5.12e8·7-s − 4.92e9·8-s − 6.58e9·9-s + 3.37e10·11-s − 3.84e11·12-s − 8.63e11·13-s + 1.49e12·14-s + 1.28e12·16-s − 1.76e13·17-s + 1.91e13·18-s + 6.52e13·19-s + 4.26e13·21-s − 9.81e13·22-s − 3.06e14·23-s + 4.09e14·24-s + 2.51e15·26-s + 2.19e15·27-s − 2.36e15·28-s − 1.18e15·29-s + 8.12e15·31-s + 6.08e15·32-s − 2.80e15·33-s + 5.14e16·34-s + ⋯ |
| L(s) = 1 | − 2.00·2-s − 0.813·3-s + 2.20·4-s + 1.63·6-s − 0.685·7-s − 1.62·8-s − 0.629·9-s + 0.392·11-s − 1.79·12-s − 1.73·13-s + 1.37·14-s + 0.292·16-s − 2.12·17-s + 1.26·18-s + 2.44·19-s + 0.558·21-s − 0.787·22-s − 1.54·23-s + 1.31·24-s + 3.49·26-s + 2.04·27-s − 1.51·28-s − 0.522·29-s + 1.78·31-s + 0.956·32-s − 0.319·33-s + 4.27·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(390625\) = \(5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.38312\times 10^{7}\) |
| Root analytic conductor: | \(8.35878\) |
| Motivic weight: | \(21\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 390625,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)\) |
Particular Values
| \(L(11)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{23}{2})\) | 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 | $C_2 \wr S_4$ | \( 1 + 1455 p T + 480715 p^{3} T^{2} + 20821915 p^{7} T^{3} + 369259299 p^{13} T^{4} + 20821915 p^{28} T^{5} + 480715 p^{45} T^{6} + 1455 p^{64} T^{7} + p^{84} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 83240 T + 4506236980 p T^{2} - 6378982435480 p^{4} T^{3} - 2439718099011562 p^{8} T^{4} - 6378982435480 p^{25} T^{5} + 4506236980 p^{43} T^{6} + 83240 p^{63} T^{7} + p^{84} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 512613800 T + 153461684091805300 p T^{2} + \)\(15\!\cdots\!00\)\( p^{3} T^{3} + \)\(27\!\cdots\!86\)\( p^{3} T^{4} + \)\(15\!\cdots\!00\)\( p^{24} T^{5} + 153461684091805300 p^{43} T^{6} + 512613800 p^{63} T^{7} + p^{84} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 33727076448 T + \)\(70\!\cdots\!28\)\( p T^{2} + \)\(19\!\cdots\!24\)\( p^{2} T^{3} + \)\(17\!\cdots\!70\)\( p^{3} T^{4} + \)\(19\!\cdots\!24\)\( p^{23} T^{5} + \)\(70\!\cdots\!28\)\( p^{43} T^{6} - 33727076448 p^{63} T^{7} + p^{84} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 66425551160 p T + \)\(67\!\cdots\!60\)\( p T^{2} + \)\(28\!\cdots\!60\)\( p^{2} T^{3} + \)\(13\!\cdots\!54\)\( p^{3} T^{4} + \)\(28\!\cdots\!60\)\( p^{23} T^{5} + \)\(67\!\cdots\!60\)\( p^{43} T^{6} + 66425551160 p^{64} T^{7} + p^{84} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 1040864182440 p T + \)\(91\!\cdots\!40\)\( p^{2} T^{2} + \)\(58\!\cdots\!20\)\( p^{3} T^{3} + \)\(34\!\cdots\!18\)\( p^{4} T^{4} + \)\(58\!\cdots\!20\)\( p^{24} T^{5} + \)\(91\!\cdots\!40\)\( p^{44} T^{6} + 1040864182440 p^{64} T^{7} + p^{84} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 3432505097360 p T + \)\(11\!\cdots\!16\)\( p^{2} T^{2} - \)\(20\!\cdots\!20\)\( p^{3} T^{3} + \)\(36\!\cdots\!46\)\( p^{4} T^{4} - \)\(20\!\cdots\!20\)\( p^{24} T^{5} + \)\(11\!\cdots\!16\)\( p^{44} T^{6} - 3432505097360 p^{64} T^{7} + p^{84} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 306130984922520 T + \)\(11\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!60\)\( T^{3} + \)\(52\!\cdots\!58\)\( T^{4} + \)\(20\!\cdots\!60\)\( p^{21} T^{5} + \)\(11\!\cdots\!20\)\( p^{42} T^{6} + 306130984922520 p^{63} T^{7} + p^{84} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 1183867881941640 T + \)\(13\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!46\)\( T^{4} + \)\(13\!\cdots\!80\)\( p^{21} T^{5} + \)\(13\!\cdots\!16\)\( p^{42} T^{6} + 1183867881941640 p^{63} T^{7} + p^{84} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 8127647981305328 T + \)\(72\!\cdots\!68\)\( T^{2} - \)\(30\!\cdots\!76\)\( T^{3} + \)\(17\!\cdots\!70\)\( T^{4} - \)\(30\!\cdots\!76\)\( p^{21} T^{5} + \)\(72\!\cdots\!68\)\( p^{42} T^{6} - 8127647981305328 p^{63} T^{7} + p^{84} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 50729262881561240 T + \)\(22\!\cdots\!80\)\( T^{2} + \)\(90\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!38\)\( T^{4} + \)\(90\!\cdots\!80\)\( p^{21} T^{5} + \)\(22\!\cdots\!80\)\( p^{42} T^{6} + 50729262881561240 p^{63} T^{7} + p^{84} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 151242912792059832 T + \)\(21\!\cdots\!48\)\( T^{2} + \)\(23\!\cdots\!84\)\( T^{3} + \)\(25\!\cdots\!70\)\( T^{4} + \)\(23\!\cdots\!84\)\( p^{21} T^{5} + \)\(21\!\cdots\!48\)\( p^{42} T^{6} + 151242912792059832 p^{63} T^{7} + p^{84} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 195241056541629800 T + \)\(55\!\cdots\!00\)\( T^{2} + \)\(87\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!98\)\( T^{4} + \)\(87\!\cdots\!00\)\( p^{21} T^{5} + \)\(55\!\cdots\!00\)\( p^{42} T^{6} + 195241056541629800 p^{63} T^{7} + p^{84} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 461921266212226680 T + \)\(47\!\cdots\!40\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(90\!\cdots\!18\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{21} T^{5} + \)\(47\!\cdots\!40\)\( p^{42} T^{6} - 461921266212226680 p^{63} T^{7} + p^{84} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 1013253578565011640 T + \)\(60\!\cdots\!40\)\( T^{2} + \)\(49\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!18\)\( T^{4} + \)\(49\!\cdots\!20\)\( p^{21} T^{5} + \)\(60\!\cdots\!40\)\( p^{42} T^{6} + 1013253578565011640 p^{63} T^{7} + p^{84} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 9994246603380823920 T + \)\(70\!\cdots\!36\)\( T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!86\)\( T^{4} - \)\(36\!\cdots\!40\)\( p^{21} T^{5} + \)\(70\!\cdots\!36\)\( p^{42} T^{6} - 9994246603380823920 p^{63} T^{7} + p^{84} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 10817434481929098952 T + \)\(10\!\cdots\!08\)\( T^{2} + \)\(81\!\cdots\!04\)\( T^{3} + \)\(49\!\cdots\!70\)\( T^{4} + \)\(81\!\cdots\!04\)\( p^{21} T^{5} + \)\(10\!\cdots\!08\)\( p^{42} T^{6} + 10817434481929098952 p^{63} T^{7} + p^{84} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 21966891437417653880 T + \)\(91\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!78\)\( T^{4} + \)\(14\!\cdots\!60\)\( p^{21} T^{5} + \)\(91\!\cdots\!60\)\( p^{42} T^{6} + 21966891437417653880 p^{63} T^{7} + p^{84} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 51295204524235287888 T + \)\(31\!\cdots\!88\)\( T^{2} - \)\(91\!\cdots\!36\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} - \)\(91\!\cdots\!36\)\( p^{21} T^{5} + \)\(31\!\cdots\!88\)\( p^{42} T^{6} - 51295204524235287888 p^{63} T^{7} + p^{84} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 74737166154487034120 T + \)\(61\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!58\)\( T^{4} + \)\(27\!\cdots\!60\)\( p^{21} T^{5} + \)\(61\!\cdots\!20\)\( p^{42} T^{6} + 74737166154487034120 p^{63} T^{7} + p^{84} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 60935090437726582240 T + \)\(11\!\cdots\!16\)\( T^{2} + \)\(77\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!46\)\( T^{4} + \)\(77\!\cdots\!80\)\( p^{21} T^{5} + \)\(11\!\cdots\!16\)\( p^{42} T^{6} + 60935090437726582240 p^{63} T^{7} + p^{84} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - \)\(27\!\cdots\!40\)\( T + \)\(87\!\cdots\!60\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!78\)\( T^{4} - \)\(14\!\cdots\!20\)\( p^{21} T^{5} + \)\(87\!\cdots\!60\)\( p^{42} T^{6} - \)\(27\!\cdots\!40\)\( p^{63} T^{7} + p^{84} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + \)\(32\!\cdots\!20\)\( T + \)\(23\!\cdots\!56\)\( T^{2} + \)\(33\!\cdots\!40\)\( T^{3} + \)\(21\!\cdots\!26\)\( T^{4} + \)\(33\!\cdots\!40\)\( p^{21} T^{5} + \)\(23\!\cdots\!56\)\( p^{42} T^{6} + \)\(32\!\cdots\!20\)\( p^{63} T^{7} + p^{84} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + \)\(30\!\cdots\!20\)\( T + \)\(61\!\cdots\!40\)\( T^{2} + \)\(58\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!18\)\( T^{4} + \)\(58\!\cdots\!40\)\( p^{21} T^{5} + \)\(61\!\cdots\!40\)\( p^{42} T^{6} + \)\(30\!\cdots\!20\)\( p^{63} T^{7} + p^{84} T^{8} \) | |
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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
−9.512919283570519278647779130129, −8.927484107686530332642340913586, −8.743497301720266241955956872533, −8.459062221790708870208194241291, −8.168514567373625497977900769585, −7.63372289677208982016064710528, −7.39965282107664482239588603424, −6.98011957666693122187570252248, −6.68380469167136614741780022838, −6.65134450631719081850348845285, −6.05646390921327101088477536275, −5.82102202973230155492226245996, −5.31484331285542439672412062362, −4.80927226859445507313629494415, −4.76111393680245235132206519734, −4.42927264509309146228516725223, −3.65065432574948832421716678225, −3.18470547250386420246729630879, −3.12462825014407143504247420090, −2.57740347695124848596849097296, −2.17697528379017191742151414164, −1.98406253829696811562023000696, −1.51324817537396040343213005539, −1.00606660954251218972629462242, −0.906350165217382717559562215399, 0, 0, 0, 0, 0.906350165217382717559562215399, 1.00606660954251218972629462242, 1.51324817537396040343213005539, 1.98406253829696811562023000696, 2.17697528379017191742151414164, 2.57740347695124848596849097296, 3.12462825014407143504247420090, 3.18470547250386420246729630879, 3.65065432574948832421716678225, 4.42927264509309146228516725223, 4.76111393680245235132206519734, 4.80927226859445507313629494415, 5.31484331285542439672412062362, 5.82102202973230155492226245996, 6.05646390921327101088477536275, 6.65134450631719081850348845285, 6.68380469167136614741780022838, 6.98011957666693122187570252248, 7.39965282107664482239588603424, 7.63372289677208982016064710528, 8.168514567373625497977900769585, 8.459062221790708870208194241291, 8.743497301720266241955956872533, 8.927484107686530332642340913586, 9.512919283570519278647779130129