Dirichlet series
| L(s) = 1 | − 64·2-s − 324·3-s + 2.56e3·4-s − 942·5-s + 2.07e4·6-s − 2.08e3·7-s − 8.19e4·8-s + 6.56e4·9-s + 6.02e4·10-s − 6.84e4·11-s − 8.29e5·12-s + 6.47e4·13-s + 1.33e5·14-s + 3.05e5·15-s + 2.29e6·16-s − 3.34e5·17-s − 4.19e6·18-s − 2.03e5·19-s − 2.41e6·20-s + 6.75e5·21-s + 4.38e6·22-s + 2.50e6·23-s + 2.65e7·24-s − 4.72e4·25-s − 4.14e6·26-s − 1.06e7·27-s − 5.33e6·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s − 0.674·5-s + 6.53·6-s − 0.328·7-s − 7.07·8-s + 10/3·9-s + 1.90·10-s − 1.40·11-s − 11.5·12-s + 0.628·13-s + 0.927·14-s + 1.55·15-s + 35/4·16-s − 0.970·17-s − 9.42·18-s − 0.358·19-s − 3.37·20-s + 0.757·21-s + 3.98·22-s + 1.86·23-s + 16.3·24-s − 0.0241·25-s − 1.77·26-s − 3.84·27-s − 1.64·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 17^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.61641\times 10^{6}\) |
| Root analytic conductor: | \(7.24801\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 17^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 2 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| 17 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 942 T + 934613 T^{2} + 1100109574 T^{3} + 847807921824 p T^{4} + 1100109574 p^{9} T^{5} + 934613 p^{18} T^{6} + 942 p^{27} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 2084 T + 59781168 T^{2} + 52072825324 p T^{3} + 44362351573342 p^{2} T^{4} + 52072825324 p^{10} T^{5} + 59781168 p^{18} T^{6} + 2084 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 68450 T + 3257428589 T^{2} + 204448796144850 T^{3} + 14027952509040213036 T^{4} + 204448796144850 p^{9} T^{5} + 3257428589 p^{18} T^{6} + 68450 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 64722 T + 20766442185 T^{2} - 1659666596793042 T^{3} + \)\(28\!\cdots\!12\)\( T^{4} - 1659666596793042 p^{9} T^{5} + 20766442185 p^{18} T^{6} - 64722 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 203854 T + 461367042397 T^{2} - 49478645428835698 T^{3} + \)\(12\!\cdots\!20\)\( T^{4} - 49478645428835698 p^{9} T^{5} + 461367042397 p^{18} T^{6} + 203854 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 2505834 T + 8900696015753 T^{2} - 13546443149992969830 T^{3} + \)\(25\!\cdots\!76\)\( T^{4} - 13546443149992969830 p^{9} T^{5} + 8900696015753 p^{18} T^{6} - 2505834 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 885976 T + 14478752317672 T^{2} - 38527592835287782792 T^{3} + \)\(58\!\cdots\!90\)\( T^{4} - 38527592835287782792 p^{9} T^{5} + 14478752317672 p^{18} T^{6} + 885976 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 11921296 T + 122043080750420 T^{2} - \)\(83\!\cdots\!84\)\( T^{3} + \)\(51\!\cdots\!14\)\( T^{4} - \)\(83\!\cdots\!84\)\( p^{9} T^{5} + 122043080750420 p^{18} T^{6} - 11921296 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 14106124 T + 97343879220004 T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!34\)\( T^{4} - \)\(11\!\cdots\!68\)\( p^{9} T^{5} + 97343879220004 p^{18} T^{6} - 14106124 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 14158146 T + 873923974921169 T^{2} + \)\(60\!\cdots\!38\)\( T^{3} + \)\(32\!\cdots\!76\)\( T^{4} + \)\(60\!\cdots\!38\)\( p^{9} T^{5} + 873923974921169 p^{18} T^{6} + 14158146 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 26846790 T + 1880657684314861 T^{2} - \)\(36\!\cdots\!02\)\( T^{3} + \)\(14\!\cdots\!40\)\( T^{4} - \)\(36\!\cdots\!02\)\( p^{9} T^{5} + 1880657684314861 p^{18} T^{6} - 26846790 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 35033332 T + 758870647147488 T^{2} - \)\(21\!\cdots\!80\)\( T^{3} - \)\(38\!\cdots\!62\)\( T^{4} - \)\(21\!\cdots\!80\)\( p^{9} T^{5} + 758870647147488 p^{18} T^{6} + 35033332 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 95616880 T + 11806223133231948 T^{2} + \)\(88\!\cdots\!16\)\( T^{3} + \)\(56\!\cdots\!42\)\( T^{4} + \)\(88\!\cdots\!16\)\( p^{9} T^{5} + 11806223133231948 p^{18} T^{6} + 95616880 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 102623436 T + 31420769428345856 T^{2} + \)\(22\!\cdots\!28\)\( T^{3} + \)\(38\!\cdots\!90\)\( T^{4} + \)\(22\!\cdots\!28\)\( p^{9} T^{5} + 31420769428345856 p^{18} T^{6} + 102623436 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 3219364 p T + 44747411705030836 T^{2} + \)\(63\!\cdots\!92\)\( T^{3} + \)\(78\!\cdots\!38\)\( T^{4} + \)\(63\!\cdots\!92\)\( p^{9} T^{5} + 44747411705030836 p^{18} T^{6} + 3219364 p^{28} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 44116808 T + 30968333032522636 T^{2} + \)\(12\!\cdots\!12\)\( T^{3} + \)\(97\!\cdots\!74\)\( T^{4} + \)\(12\!\cdots\!12\)\( p^{9} T^{5} + 30968333032522636 p^{18} T^{6} - 44116808 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 391106100 T + 182111416980969488 T^{2} + \)\(45\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!46\)\( T^{4} + \)\(45\!\cdots\!52\)\( p^{9} T^{5} + 182111416980969488 p^{18} T^{6} + 391106100 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 325868288 T + 165923758232723020 T^{2} + \)\(42\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} + \)\(42\!\cdots\!52\)\( p^{9} T^{5} + 165923758232723020 p^{18} T^{6} + 325868288 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 108839048 T + 339680498966230036 T^{2} - \)\(32\!\cdots\!96\)\( T^{3} + \)\(57\!\cdots\!46\)\( T^{4} - \)\(32\!\cdots\!96\)\( p^{9} T^{5} + 339680498966230036 p^{18} T^{6} - 108839048 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 319973836 T - 7247970805548976 T^{2} + \)\(24\!\cdots\!32\)\( T^{3} + \)\(56\!\cdots\!66\)\( T^{4} + \)\(24\!\cdots\!32\)\( p^{9} T^{5} - 7247970805548976 p^{18} T^{6} + 319973836 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 829727740 T + 927405637227227112 T^{2} + \)\(71\!\cdots\!96\)\( T^{3} + \)\(43\!\cdots\!86\)\( T^{4} + \)\(71\!\cdots\!96\)\( p^{9} T^{5} + 927405637227227112 p^{18} T^{6} + 829727740 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 354724644 T + 1111088532256145400 T^{2} - \)\(86\!\cdots\!56\)\( T^{3} + \)\(20\!\cdots\!70\)\( T^{4} - \)\(86\!\cdots\!56\)\( p^{9} T^{5} + 1111088532256145400 p^{18} T^{6} + 354724644 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−9.196396598252049435979873647852, −8.337799627954032389161844762638, −8.261530313733196018054798451309, −8.250275838249398113155912208417, −7.88421010892811259258415184018, −7.50741046084711803522807878184, −7.07868877396585534919791095978, −6.89872115969292331187288382460, −6.77717241702754733219902159105, −6.41313768170534137203718219760, −6.05448519454466685026424899442, −5.82250421219577684063081591918, −5.64250935847980790234855822434, −5.01864273838337973347320873361, −4.64153763055565023815474297948, −4.54153847122652392853115813656, −4.09404730306663039499949481892, −3.31980134273535418419239643092, −2.96854089620811155564871705188, −2.73171179649299742043886149255, −2.40026326711956703370351588262, −1.63307185839631150011849960957, −1.34983900524475850861280788490, −1.13149895360799588688819946724, −0.945738980675199760746925605121, 0, 0, 0, 0, 0.945738980675199760746925605121, 1.13149895360799588688819946724, 1.34983900524475850861280788490, 1.63307185839631150011849960957, 2.40026326711956703370351588262, 2.73171179649299742043886149255, 2.96854089620811155564871705188, 3.31980134273535418419239643092, 4.09404730306663039499949481892, 4.54153847122652392853115813656, 4.64153763055565023815474297948, 5.01864273838337973347320873361, 5.64250935847980790234855822434, 5.82250421219577684063081591918, 6.05448519454466685026424899442, 6.41313768170534137203718219760, 6.77717241702754733219902159105, 6.89872115969292331187288382460, 7.07868877396585534919791095978, 7.50741046084711803522807878184, 7.88421010892811259258415184018, 8.250275838249398113155912208417, 8.261530313733196018054798451309, 8.337799627954032389161844762638, 9.196396598252049435979873647852