Dirichlet series
| L(s) = 1 | + 64·2-s − 324·3-s + 2.56e3·4-s − 2.50e3·5-s − 2.07e4·6-s − 7.10e3·7-s + 8.19e4·8-s + 6.56e4·9-s − 1.60e5·10-s − 8.24e4·11-s − 8.29e5·12-s + 1.14e5·13-s − 4.54e5·14-s + 8.10e5·15-s + 2.29e6·16-s − 1.08e5·17-s + 4.19e6·18-s + 2.96e5·19-s − 6.40e6·20-s + 2.30e6·21-s − 5.27e6·22-s + 8.22e5·23-s − 2.65e7·24-s + 3.90e6·25-s + 7.31e6·26-s − 1.06e7·27-s − 1.81e7·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s − 6.53·6-s − 1.11·7-s + 7.07·8-s + 10/3·9-s − 5.05·10-s − 1.69·11-s − 11.5·12-s + 1.10·13-s − 3.16·14-s + 4.13·15-s + 35/4·16-s − 0.315·17-s + 9.42·18-s + 0.521·19-s − 8.94·20-s + 2.58·21-s − 4.80·22-s + 0.612·23-s − 16.3·24-s + 2·25-s + 3.13·26-s − 3.84·27-s − 5.59·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.62782\times 10^{9}\) |
| Root analytic conductor: | \(14.1726\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| 5 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 145 p^{2} T + 2112778 p^{2} T^{2} + 257152165 p^{4} T^{3} + 2509785580810 p^{4} T^{4} + 257152165 p^{13} T^{5} + 2112778 p^{20} T^{6} + 145 p^{29} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 82455 T + 11231675396 T^{2} + 585691632554823 T^{3} + 41677721607542949990 T^{4} + 585691632554823 p^{9} T^{5} + 11231675396 p^{18} T^{6} + 82455 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 108729 T + 368384206076 T^{2} + 49111919155642959 T^{3} + \)\(59\!\cdots\!26\)\( T^{4} + 49111919155642959 p^{9} T^{5} + 368384206076 p^{18} T^{6} + 108729 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 296228 T + 720107985232 T^{2} - 114825747543439268 T^{3} + \)\(25\!\cdots\!42\)\( T^{4} - 114825747543439268 p^{9} T^{5} + 720107985232 p^{18} T^{6} - 296228 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 822183 T + 480444841538 T^{2} - 314203130133468579 T^{3} + \)\(10\!\cdots\!38\)\( T^{4} - 314203130133468579 p^{9} T^{5} + 480444841538 p^{18} T^{6} - 822183 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 1975566 T + 3444045755696 T^{2} - 52259338621619121942 T^{3} - \)\(15\!\cdots\!90\)\( T^{4} - 52259338621619121942 p^{9} T^{5} + 3444045755696 p^{18} T^{6} + 1975566 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 5589746 T + 114776805428356 T^{2} - \)\(44\!\cdots\!22\)\( T^{3} + \)\(46\!\cdots\!02\)\( T^{4} - \)\(44\!\cdots\!22\)\( p^{9} T^{5} + 114776805428356 p^{18} T^{6} - 5589746 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 16320887 T + 406930522684510 T^{2} - \)\(40\!\cdots\!37\)\( T^{3} + \)\(66\!\cdots\!82\)\( T^{4} - \)\(40\!\cdots\!37\)\( p^{9} T^{5} + 406930522684510 p^{18} T^{6} - 16320887 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 26202771 T + 582692241488750 T^{2} - \)\(16\!\cdots\!29\)\( p T^{3} + \)\(15\!\cdots\!34\)\( T^{4} - \)\(16\!\cdots\!29\)\( p^{10} T^{5} + 582692241488750 p^{18} T^{6} - 26202771 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 28277132 T + 2050023954952156 T^{2} - \)\(40\!\cdots\!76\)\( T^{3} + \)\(15\!\cdots\!14\)\( T^{4} - \)\(40\!\cdots\!76\)\( p^{9} T^{5} + 2050023954952156 p^{18} T^{6} - 28277132 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 33897426 T + 1942480379469500 T^{2} + \)\(11\!\cdots\!74\)\( T^{3} + \)\(92\!\cdots\!82\)\( T^{4} + \)\(11\!\cdots\!74\)\( p^{9} T^{5} + 1942480379469500 p^{18} T^{6} + 33897426 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 29406027 T - 1986661712549782 T^{2} + \)\(62\!\cdots\!33\)\( T^{3} + \)\(20\!\cdots\!10\)\( T^{4} + \)\(62\!\cdots\!33\)\( p^{9} T^{5} - 1986661712549782 p^{18} T^{6} + 29406027 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 77844624 T + 31147371792212156 T^{2} + \)\(18\!\cdots\!08\)\( T^{3} + \)\(38\!\cdots\!26\)\( T^{4} + \)\(18\!\cdots\!08\)\( p^{9} T^{5} + 31147371792212156 p^{18} T^{6} + 77844624 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 166002143 T + 49230256566299530 T^{2} - \)\(57\!\cdots\!29\)\( T^{3} + \)\(87\!\cdots\!98\)\( T^{4} - \)\(57\!\cdots\!29\)\( p^{9} T^{5} + 49230256566299530 p^{18} T^{6} - 166002143 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 38512556 T + 52486672128432364 T^{2} - \)\(31\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!14\)\( T^{4} - \)\(31\!\cdots\!72\)\( p^{9} T^{5} + 52486672128432364 p^{18} T^{6} - 38512556 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 125727621 T + 167839053352692308 T^{2} - \)\(16\!\cdots\!89\)\( T^{3} + \)\(11\!\cdots\!62\)\( T^{4} - \)\(16\!\cdots\!89\)\( p^{9} T^{5} + 167839053352692308 p^{18} T^{6} - 125727621 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 1107146 p T + 225111966385141120 T^{2} - \)\(13\!\cdots\!78\)\( T^{3} + \)\(19\!\cdots\!62\)\( T^{4} - \)\(13\!\cdots\!78\)\( p^{9} T^{5} + 225111966385141120 p^{18} T^{6} - 1107146 p^{28} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 80878901 T + 306903884452726324 T^{2} - \)\(33\!\cdots\!09\)\( T^{3} + \)\(47\!\cdots\!94\)\( T^{4} - \)\(33\!\cdots\!09\)\( p^{9} T^{5} + 306903884452726324 p^{18} T^{6} - 80878901 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 436233516 T + 507813292629029564 T^{2} + \)\(17\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!14\)\( T^{4} + \)\(17\!\cdots\!04\)\( p^{9} T^{5} + 507813292629029564 p^{18} T^{6} + 436233516 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 427530087 T + 886198547349172958 T^{2} + \)\(29\!\cdots\!29\)\( T^{3} + \)\(37\!\cdots\!82\)\( T^{4} + \)\(29\!\cdots\!29\)\( p^{9} T^{5} + 886198547349172958 p^{18} T^{6} + 427530087 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 239018831 T + 2701354425810304756 T^{2} - \)\(50\!\cdots\!93\)\( T^{3} + \)\(29\!\cdots\!30\)\( T^{4} - \)\(50\!\cdots\!93\)\( p^{9} T^{5} + 2701354425810304756 p^{18} T^{6} - 239018831 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.03364321460722895175836001775, −6.54159927301356085469610382809, −6.45189390177319702290427035393, −6.35734287963711630700683539340, −6.23932368602596600052906555535, −5.58034137148619514713475236132, −5.52679345790101896269768632239, −5.50523568099406203793226224885, −5.33269723817805031873936837645, −4.66069413915680044686579218536, −4.56909651987946213645257842269, −4.51227109760598719096059490798, −4.48864140102081997333424925022, −3.75991682291886132657379282298, −3.64030415467393308378608846076, −3.59744687484360748502206429419, −3.40117038901776694405051330980, −2.75700732740611469680524773849, −2.58817255390636706336668007544, −2.44059144589305527800117757356, −2.34838796287396130159883951023, −1.28111362022745490077950681896, −1.22205965731733420928718893032, −1.11135796438829414138298009321, −1.03959590058932724847539898346, 0, 0, 0, 0, 1.03959590058932724847539898346, 1.11135796438829414138298009321, 1.22205965731733420928718893032, 1.28111362022745490077950681896, 2.34838796287396130159883951023, 2.44059144589305527800117757356, 2.58817255390636706336668007544, 2.75700732740611469680524773849, 3.40117038901776694405051330980, 3.59744687484360748502206429419, 3.64030415467393308378608846076, 3.75991682291886132657379282298, 4.48864140102081997333424925022, 4.51227109760598719096059490798, 4.56909651987946213645257842269, 4.66069413915680044686579218536, 5.33269723817805031873936837645, 5.50523568099406203793226224885, 5.52679345790101896269768632239, 5.58034137148619514713475236132, 6.23932368602596600052906555535, 6.35734287963711630700683539340, 6.45189390177319702290427035393, 6.54159927301356085469610382809, 7.03364321460722895175836001775