Dirichlet series
| L(s) = 1 | + 64·2-s − 324·3-s + 2.56e3·4-s + 2.50e3·5-s − 2.07e4·6-s + 4.58e3·7-s + 8.19e4·8-s + 6.56e4·9-s + 1.60e5·10-s − 3.28e4·11-s − 8.29e5·12-s − 1.14e5·13-s + 2.93e5·14-s − 8.10e5·15-s + 2.29e6·16-s + 3.60e5·17-s + 4.19e6·18-s − 5.07e5·19-s + 6.40e6·20-s − 1.48e6·21-s − 2.10e6·22-s − 2.13e6·23-s − 2.65e7·24-s + 3.90e6·25-s − 7.31e6·26-s − 1.06e7·27-s + 1.17e7·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s + 0.721·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s − 0.676·11-s − 11.5·12-s − 1.10·13-s + 2.04·14-s − 4.13·15-s + 35/4·16-s + 1.04·17-s + 9.42·18-s − 0.893·19-s + 8.94·20-s − 1.66·21-s − 1.91·22-s − 1.59·23-s − 16.3·24-s + 2·25-s − 3.13·26-s − 3.84·27-s + 3.60·28-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 | $C_2 \wr S_4$ | \( 1 - 655 p T + 18665786 p T^{2} - 11247493237 p^{2} T^{3} + 21439428583694 p^{3} T^{4} - 11247493237 p^{11} T^{5} + 18665786 p^{19} T^{6} - 655 p^{28} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2985 p T + 3743036574 T^{2} + 173346754904355 T^{3} + 12710421339906478306 T^{4} + 173346754904355 p^{9} T^{5} + 3743036574 p^{18} T^{6} + 2985 p^{28} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 360625 T + 278583863078 T^{2} - 69873907391019175 T^{3} + \)\(41\!\cdots\!14\)\( T^{4} - 69873907391019175 p^{9} T^{5} + 278583863078 p^{18} T^{6} - 360625 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 507548 T + 700017311816 T^{2} + 344008995879897596 T^{3} + \)\(24\!\cdots\!46\)\( T^{4} + 344008995879897596 p^{9} T^{5} + 700017311816 p^{18} T^{6} + 507548 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 2139319 T + 5018901140232 T^{2} + 4577172448195713051 T^{3} + \)\(81\!\cdots\!94\)\( T^{4} + 4577172448195713051 p^{9} T^{5} + 5018901140232 p^{18} T^{6} + 2139319 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 7840156 T + 48512766172588 T^{2} + \)\(23\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!82\)\( T^{4} + \)\(23\!\cdots\!80\)\( p^{9} T^{5} + 48512766172588 p^{18} T^{6} + 7840156 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 258936 T + 33346581924616 T^{2} + 86835708710033955272 T^{3} + \)\(10\!\cdots\!78\)\( T^{4} + 86835708710033955272 p^{9} T^{5} + 33346581924616 p^{18} T^{6} - 258936 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 4604671 T + 155351809628378 T^{2} + \)\(17\!\cdots\!97\)\( T^{3} + \)\(20\!\cdots\!98\)\( T^{4} + \)\(17\!\cdots\!97\)\( p^{9} T^{5} + 155351809628378 p^{18} T^{6} + 4604671 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 30020835 T + 1469921241923790 T^{2} + \)\(29\!\cdots\!21\)\( T^{3} + \)\(74\!\cdots\!50\)\( T^{4} + \)\(29\!\cdots\!21\)\( p^{9} T^{5} + 1469921241923790 p^{18} T^{6} + 30020835 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 39984692 T + 711052878429340 T^{2} + \)\(25\!\cdots\!08\)\( T^{3} + \)\(90\!\cdots\!70\)\( T^{4} + \)\(25\!\cdots\!08\)\( p^{9} T^{5} + 711052878429340 p^{18} T^{6} + 39984692 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 59831272 T + 3213959675178696 T^{2} + \)\(10\!\cdots\!36\)\( T^{3} + \)\(40\!\cdots\!46\)\( T^{4} + \)\(10\!\cdots\!36\)\( p^{9} T^{5} + 3213959675178696 p^{18} T^{6} + 59831272 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 1460571 T + 8591412233751198 T^{2} + \)\(11\!\cdots\!05\)\( T^{3} + \)\(38\!\cdots\!26\)\( T^{4} + \)\(11\!\cdots\!05\)\( p^{9} T^{5} + 8591412233751198 p^{18} T^{6} + 1460571 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 162203918 T + 32143448611049884 T^{2} - \)\(29\!\cdots\!02\)\( T^{3} + \)\(36\!\cdots\!70\)\( T^{4} - \)\(29\!\cdots\!02\)\( p^{9} T^{5} + 32143448611049884 p^{18} T^{6} - 162203918 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 65002583 T + 31556634543740002 T^{2} - \)\(19\!\cdots\!69\)\( T^{3} + \)\(50\!\cdots\!50\)\( T^{4} - \)\(19\!\cdots\!69\)\( p^{9} T^{5} + 31556634543740002 p^{18} T^{6} - 65002583 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 4589694 T + 61566016868333668 T^{2} - \)\(22\!\cdots\!54\)\( T^{3} + \)\(21\!\cdots\!74\)\( T^{4} - \)\(22\!\cdots\!54\)\( p^{9} T^{5} + 61566016868333668 p^{18} T^{6} - 4589694 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 36815019 T + 159282448286985626 T^{2} + \)\(57\!\cdots\!19\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(57\!\cdots\!19\)\( p^{9} T^{5} + 159282448286985626 p^{18} T^{6} + 36815019 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 23733856 T + 178886014677615268 T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(15\!\cdots\!80\)\( p^{9} T^{5} + 178886014677615268 p^{18} T^{6} - 23733856 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 117872777 T + 365807282444413936 T^{2} - \)\(56\!\cdots\!57\)\( T^{3} + \)\(58\!\cdots\!30\)\( T^{4} - \)\(56\!\cdots\!57\)\( p^{9} T^{5} + 365807282444413936 p^{18} T^{6} - 117872777 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 235800270 T + 433196274151057796 T^{2} + \)\(88\!\cdots\!58\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} + \)\(88\!\cdots\!58\)\( p^{9} T^{5} + 433196274151057796 p^{18} T^{6} + 235800270 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 493622013 T + 959702610883999626 T^{2} + \)\(32\!\cdots\!23\)\( T^{3} + \)\(41\!\cdots\!30\)\( T^{4} + \)\(32\!\cdots\!23\)\( p^{9} T^{5} + 959702610883999626 p^{18} T^{6} + 493622013 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 325597225 T + 2785341192642361574 T^{2} + \)\(65\!\cdots\!99\)\( T^{3} + \)\(30\!\cdots\!50\)\( T^{4} + \)\(65\!\cdots\!99\)\( p^{9} T^{5} + 2785341192642361574 p^{18} T^{6} + 325597225 p^{27} T^{7} + p^{36} T^{8} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12490293030069759626136487923, −6.41577442343778187096062578856, −6.38334219525291709988439045144, −6.23498306156110675591339878558, −6.22305173451837493235422089772, −5.75204828403392047362785331110, −5.43546221947567511563173663239, −5.38772563594061249596344746380, −5.25672315991773754715518790887, −4.90569190237214338778341592724, −4.84951598538879799646627632808, −4.66698734706667877601295197842, −4.45126108273533805145665053291, −3.74405764070323285929966810955, −3.69710537926375970744790525575, −3.49564642475986972808620628770, −3.39444747728223401646997700439, −2.53208071446648687713463988017, −2.40156182851564056979367621446, −2.35058454226828586100237991314, −2.07911798401449982510309312566, −1.55859786262443031391384941743, −1.32669975838876546511400449593, −1.32634481440224686486656918680, −1.25210319989852239061082119057, 0, 0, 0, 0, 1.25210319989852239061082119057, 1.32634481440224686486656918680, 1.32669975838876546511400449593, 1.55859786262443031391384941743, 2.07911798401449982510309312566, 2.35058454226828586100237991314, 2.40156182851564056979367621446, 2.53208071446648687713463988017, 3.39444747728223401646997700439, 3.49564642475986972808620628770, 3.69710537926375970744790525575, 3.74405764070323285929966810955, 4.45126108273533805145665053291, 4.66698734706667877601295197842, 4.84951598538879799646627632808, 4.90569190237214338778341592724, 5.25672315991773754715518790887, 5.38772563594061249596344746380, 5.43546221947567511563173663239, 5.75204828403392047362785331110, 6.22305173451837493235422089772, 6.23498306156110675591339878558, 6.38334219525291709988439045144, 6.41577442343778187096062578856, 7.12490293030069759626136487923