Dirichlet series
| L(s) = 1 | + 3·2-s + 324·3-s − 721·4-s + 972·6-s − 9.83e3·7-s − 5.24e3·8-s + 6.56e4·9-s − 3.59e4·11-s − 2.33e5·12-s − 7.99e4·13-s − 2.95e4·14-s − 1.39e4·16-s − 6.67e5·17-s + 1.96e5·18-s − 4.25e5·19-s − 3.18e6·21-s − 1.07e5·22-s − 1.03e6·23-s − 1.69e6·24-s − 2.39e5·26-s + 1.06e7·27-s + 7.09e6·28-s + 3.67e4·29-s + 2.37e5·31-s + 2.40e6·32-s − 1.16e7·33-s − 2.00e6·34-s + ⋯ |
| L(s) = 1 | + 0.132·2-s + 2.30·3-s − 1.40·4-s + 0.306·6-s − 1.54·7-s − 0.452·8-s + 10/3·9-s − 0.741·11-s − 3.25·12-s − 0.776·13-s − 0.205·14-s − 0.0530·16-s − 1.93·17-s + 0.441·18-s − 0.749·19-s − 3.57·21-s − 0.0982·22-s − 0.768·23-s − 1.04·24-s − 0.102·26-s + 3.84·27-s + 2.17·28-s + 0.00965·29-s + 0.0461·31-s + 0.405·32-s − 1.71·33-s − 0.257·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(31640625\) = \(3^{4} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.22635\times 10^{6}\) |
| Root analytic conductor: | \(6.21511\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 31640625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 5 | \( 1 \) | ||
| good | 2 | $C_2 \wr S_4$ | \( 1 - 3 T + 365 p T^{2} + 111 p^{3} T^{3} + 4077 p^{7} T^{4} + 111 p^{12} T^{5} + 365 p^{19} T^{6} - 3 p^{27} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 9834 T + 141893392 T^{2} + 154775191302 p T^{3} + 167640892834110 p^{2} T^{4} + 154775191302 p^{10} T^{5} + 141893392 p^{18} T^{6} + 9834 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} + 186668393765490 p^{9} T^{5} + 5523704672 p^{18} T^{6} + 35994 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 79998 T + 31802195344 T^{2} + 1495755960326586 T^{3} + \)\(42\!\cdots\!10\)\( T^{4} + 1495755960326586 p^{9} T^{5} + 31802195344 p^{18} T^{6} + 79998 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 667878 T + 282367700632 T^{2} + 126324742595722170 T^{3} + \)\(53\!\cdots\!66\)\( T^{4} + 126324742595722170 p^{9} T^{5} + 282367700632 p^{18} T^{6} + 667878 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 425792 T + 640639588492 T^{2} + 117316391220409664 T^{3} + \)\(17\!\cdots\!54\)\( T^{4} + 117316391220409664 p^{9} T^{5} + 640639588492 p^{18} T^{6} + 425792 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 1031232 T + 5668189166908 T^{2} + 5468168853855766848 T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + 5468168853855766848 p^{9} T^{5} + 5668189166908 p^{18} T^{6} + 1031232 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 36786 T + 11698955610980 T^{2} + 38232367405598309058 T^{3} + \)\(21\!\cdots\!18\)\( T^{4} + 38232367405598309058 p^{9} T^{5} + 11698955610980 p^{18} T^{6} - 36786 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 237044 T + 50199397014268 T^{2} - \)\(19\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!72\)\( p^{9} T^{5} + 50199397014268 p^{18} T^{6} - 237044 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 15750594 T + 436368456033952 T^{2} + \)\(39\!\cdots\!34\)\( T^{3} + \)\(71\!\cdots\!50\)\( T^{4} + \)\(39\!\cdots\!34\)\( p^{9} T^{5} + 436368456033952 p^{18} T^{6} + 15750594 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 46660044 T + 1629598509234068 T^{2} - \)\(39\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!54\)\( T^{4} - \)\(39\!\cdots\!32\)\( p^{9} T^{5} + 1629598509234068 p^{18} T^{6} - 46660044 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 67170720 T + 2900332889881420 T^{2} + \)\(87\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!98\)\( T^{4} + \)\(87\!\cdots\!60\)\( p^{9} T^{5} + 2900332889881420 p^{18} T^{6} + 67170720 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 48243420 T + 3581950587681820 T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!78\)\( T^{4} + \)\(11\!\cdots\!40\)\( p^{9} T^{5} + 3581950587681820 p^{18} T^{6} + 48243420 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 198376482 T + 21847367650018168 T^{2} + \)\(16\!\cdots\!98\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(16\!\cdots\!98\)\( p^{9} T^{5} + 21847367650018168 p^{18} T^{6} + 198376482 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 118263018 T + 24386862278408432 T^{2} + \)\(17\!\cdots\!86\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} + \)\(17\!\cdots\!86\)\( p^{9} T^{5} + 24386862278408432 p^{18} T^{6} + 118263018 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 178713880 T + 47272829372304076 T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!06\)\( T^{4} + \)\(59\!\cdots\!20\)\( p^{9} T^{5} + 47272829372304076 p^{18} T^{6} + 178713880 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 16141548 T + 64587942934639852 T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!06\)\( T^{4} + \)\(16\!\cdots\!80\)\( p^{9} T^{5} + 64587942934639852 p^{18} T^{6} + 16141548 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 78445332 T + 132742682733898508 T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(78\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!24\)\( p^{9} T^{5} + 132742682733898508 p^{18} T^{6} + 78445332 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 514053252 T + 188520765370005748 T^{2} + \)\(34\!\cdots\!48\)\( T^{3} + \)\(82\!\cdots\!10\)\( T^{4} + \)\(34\!\cdots\!48\)\( p^{9} T^{5} + 188520765370005748 p^{18} T^{6} + 514053252 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 431961140 T + 452072920533003676 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!66\)\( T^{4} + \)\(14\!\cdots\!80\)\( p^{9} T^{5} + 452072920533003676 p^{18} T^{6} + 431961140 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 557494176 T + 646005074335691500 T^{2} + \)\(30\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!66\)\( T^{4} + \)\(30\!\cdots\!56\)\( p^{9} T^{5} + 646005074335691500 p^{18} T^{6} + 557494176 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 178691112 T + 708605882924008892 T^{2} + \)\(39\!\cdots\!44\)\( T^{3} + \)\(23\!\cdots\!94\)\( T^{4} + \)\(39\!\cdots\!44\)\( p^{9} T^{5} + 708605882924008892 p^{18} T^{6} + 178691112 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 840904752 T + 1010444926973516932 T^{2} - \)\(20\!\cdots\!40\)\( T^{3} + \)\(58\!\cdots\!46\)\( T^{4} - \)\(20\!\cdots\!40\)\( p^{9} T^{5} + 1010444926973516932 p^{18} T^{6} - 840904752 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−9.564601838297810711794483648673, −8.930410377440872499836593368944, −8.866651963828645397522256864310, −8.596561347885453827436033724656, −8.553286756614399690630129526038, −7.938128703253869421622348298933, −7.70647888253369870188036203133, −7.54233768811465345101971763819, −6.86734392139556086085705187368, −6.82116706498463242270679211957, −6.31762571012494607582891647668, −6.19542195966916263243401811306, −5.72542528027685596045160673313, −4.74067576741076782942105997607, −4.71651668134833929865077104985, −4.59904459873906248275581035880, −4.43788012142181119942328451025, −3.51913149261657940519071917889, −3.45017035636367489990902106310, −3.26811832345567053259374111203, −2.77744284691475268649828881811, −2.38487397252484280222985168598, −2.15669396835982072154067108896, −1.49047984582795746135900628536, −1.45518994833629041348781448288, 0, 0, 0, 0, 1.45518994833629041348781448288, 1.49047984582795746135900628536, 2.15669396835982072154067108896, 2.38487397252484280222985168598, 2.77744284691475268649828881811, 3.26811832345567053259374111203, 3.45017035636367489990902106310, 3.51913149261657940519071917889, 4.43788012142181119942328451025, 4.59904459873906248275581035880, 4.71651668134833929865077104985, 4.74067576741076782942105997607, 5.72542528027685596045160673313, 6.19542195966916263243401811306, 6.31762571012494607582891647668, 6.82116706498463242270679211957, 6.86734392139556086085705187368, 7.54233768811465345101971763819, 7.70647888253369870188036203133, 7.938128703253869421622348298933, 8.553286756614399690630129526038, 8.596561347885453827436033724656, 8.866651963828645397522256864310, 8.930410377440872499836593368944, 9.564601838297810711794483648673