Dirichlet series
| L(s) = 1 | + 5.15e10·2-s − 2.35e16·3-s + 1.77e21·4-s − 5.87e23·5-s − 1.21e27·6-s + 9.28e28·7-s + 5.07e31·8-s − 7.92e32·9-s − 3.02e34·10-s − 5.09e35·11-s − 4.17e37·12-s + 7.83e36·13-s + 4.78e39·14-s + 1.38e40·15-s + 1.30e42·16-s + 4.96e42·17-s − 4.08e43·18-s − 1.84e44·19-s − 1.04e45·20-s − 2.19e45·21-s − 2.62e46·22-s + 3.59e46·23-s − 1.19e48·24-s − 2.05e48·25-s + 4.03e47·26-s + 3.20e49·27-s + 1.64e50·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.816·3-s + 3·4-s − 0.451·5-s − 1.73·6-s + 0.648·7-s + 3.53·8-s − 0.949·9-s − 0.957·10-s − 0.601·11-s − 2.45·12-s + 0.0290·13-s + 1.37·14-s + 0.368·15-s + 15/4·16-s + 1.76·17-s − 2.01·18-s − 1.41·19-s − 1.35·20-s − 0.529·21-s − 1.27·22-s + 0.377·23-s − 2.88·24-s − 1.21·25-s + 0.0616·26-s + 1.32·27-s + 1.94·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(70-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+69/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(219288.\) |
| Root analytic conductor: | \(7.76550\) |
| Motivic weight: | \(69\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 8,\ (\ :69/2, 69/2, 69/2),\ 1)\) |
Particular Values
| \(L(35)\) | \(\approx\) | \(7.900032351\) |
| \(L(\frac12)\) | \(\approx\) | \(7.900032351\) |
| \(L(\frac{71}{2})\) | 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 | $C_1$ | \( ( 1 - p^{34} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 7866377652201484 p T + \)\(20\!\cdots\!97\)\( p^{8} T^{2} + \)\(59\!\cdots\!04\)\( p^{22} T^{3} + \)\(20\!\cdots\!97\)\( p^{77} T^{4} + 7866377652201484 p^{139} T^{5} + p^{207} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + \)\(46\!\cdots\!58\)\( p^{3} T + \)\(61\!\cdots\!27\)\( p^{8} T^{2} + \)\(16\!\cdots\!04\)\( p^{19} T^{3} + \)\(61\!\cdots\!27\)\( p^{77} T^{4} + \)\(46\!\cdots\!58\)\( p^{141} T^{5} + p^{207} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 - \)\(13\!\cdots\!88\)\( p T + \)\(21\!\cdots\!77\)\( p^{6} T^{2} - \)\(45\!\cdots\!28\)\( p^{14} T^{3} + \)\(21\!\cdots\!77\)\( p^{75} T^{4} - \)\(13\!\cdots\!88\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(50\!\cdots\!84\)\( T + \)\(13\!\cdots\!75\)\( p^{3} T^{2} + \)\(32\!\cdots\!40\)\( p^{7} T^{3} + \)\(13\!\cdots\!75\)\( p^{72} T^{4} + \)\(50\!\cdots\!84\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(60\!\cdots\!06\)\( p T + \)\(13\!\cdots\!27\)\( p^{4} T^{2} + \)\(86\!\cdots\!16\)\( p^{8} T^{3} + \)\(13\!\cdots\!27\)\( p^{73} T^{4} - \)\(60\!\cdots\!06\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(29\!\cdots\!78\)\( p T + \)\(58\!\cdots\!91\)\( p^{3} T^{2} - \)\(56\!\cdots\!76\)\( p^{5} T^{3} + \)\(58\!\cdots\!91\)\( p^{72} T^{4} - \)\(29\!\cdots\!78\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(21\!\cdots\!23\)\( p T^{2} + \)\(32\!\cdots\!00\)\( p^{4} T^{3} + \)\(21\!\cdots\!23\)\( p^{70} T^{4} + \)\(18\!\cdots\!00\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!16\)\( p T + \)\(36\!\cdots\!93\)\( p^{2} T^{2} - \)\(79\!\cdots\!52\)\( p^{3} T^{3} + \)\(36\!\cdots\!93\)\( p^{71} T^{4} - \)\(15\!\cdots\!16\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(77\!\cdots\!70\)\( T + \)\(49\!\cdots\!27\)\( p^{2} T^{2} - \)\(18\!\cdots\!60\)\( p^{4} T^{3} + \)\(49\!\cdots\!27\)\( p^{71} T^{4} - \)\(77\!\cdots\!70\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(53\!\cdots\!96\)\( T + \)\(11\!\cdots\!85\)\( p^{2} T^{2} - \)\(25\!\cdots\!00\)\( p^{4} T^{3} + \)\(11\!\cdots\!85\)\( p^{71} T^{4} - \)\(53\!\cdots\!96\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(22\!\cdots\!94\)\( T + \)\(10\!\cdots\!39\)\( p T^{2} + \)\(27\!\cdots\!72\)\( p^{2} T^{3} + \)\(10\!\cdots\!39\)\( p^{70} T^{4} + \)\(22\!\cdots\!94\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(40\!\cdots\!94\)\( T + \)\(34\!\cdots\!95\)\( p T^{2} - \)\(23\!\cdots\!40\)\( p^{2} T^{3} + \)\(34\!\cdots\!95\)\( p^{70} T^{4} + \)\(40\!\cdots\!94\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!72\)\( T + \)\(25\!\cdots\!99\)\( p T^{2} + \)\(63\!\cdots\!84\)\( p^{2} T^{3} + \)\(25\!\cdots\!99\)\( p^{70} T^{4} + \)\(13\!\cdots\!72\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!08\)\( p T + \)\(37\!\cdots\!77\)\( p^{2} T^{2} - \)\(33\!\cdots\!64\)\( p^{3} T^{3} + \)\(37\!\cdots\!77\)\( p^{71} T^{4} - \)\(15\!\cdots\!08\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(69\!\cdots\!94\)\( p T + \)\(75\!\cdots\!23\)\( p^{2} T^{2} + \)\(38\!\cdots\!48\)\( p^{3} T^{3} + \)\(75\!\cdots\!23\)\( p^{71} T^{4} + \)\(69\!\cdots\!94\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(65\!\cdots\!60\)\( p T + \)\(43\!\cdots\!23\)\( p^{3} T^{2} - \)\(65\!\cdots\!80\)\( p^{3} T^{3} + \)\(43\!\cdots\!23\)\( p^{72} T^{4} - \)\(65\!\cdots\!60\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 - \)\(69\!\cdots\!26\)\( p T + \)\(11\!\cdots\!55\)\( p^{2} T^{2} - \)\(51\!\cdots\!80\)\( p^{3} T^{3} + \)\(11\!\cdots\!55\)\( p^{71} T^{4} - \)\(69\!\cdots\!26\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 - \)\(44\!\cdots\!08\)\( p T + \)\(11\!\cdots\!57\)\( p^{2} T^{2} - \)\(20\!\cdots\!24\)\( p^{3} T^{3} + \)\(11\!\cdots\!57\)\( p^{71} T^{4} - \)\(44\!\cdots\!08\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 - \)\(26\!\cdots\!56\)\( T + \)\(39\!\cdots\!05\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(39\!\cdots\!05\)\( p^{69} T^{4} - \)\(26\!\cdots\!56\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(57\!\cdots\!62\)\( T + \)\(80\!\cdots\!87\)\( T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(80\!\cdots\!87\)\( p^{69} T^{4} + \)\(57\!\cdots\!62\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(43\!\cdots\!57\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!57\)\( p^{69} T^{4} + \)\(14\!\cdots\!20\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 + \)\(36\!\cdots\!92\)\( T + \)\(94\!\cdots\!97\)\( T^{2} + \)\(20\!\cdots\!12\)\( p T^{3} + \)\(94\!\cdots\!97\)\( p^{69} T^{4} + \)\(36\!\cdots\!92\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(17\!\cdots\!70\)\( T + \)\(96\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(96\!\cdots\!27\)\( p^{69} T^{4} - \)\(17\!\cdots\!70\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(10\!\cdots\!14\)\( T + \)\(26\!\cdots\!83\)\( T^{2} + \)\(17\!\cdots\!48\)\( T^{3} + \)\(26\!\cdots\!83\)\( p^{69} T^{4} + \)\(10\!\cdots\!14\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.39139236754961950827826358073, −11.90955870793936266870395636240, −11.62117347244427205983498729917, −11.25768378476704273024548085515, −10.67774157048055848572592132156, −10.26255468488312242667689075613, −9.744398851690347407339989222572, −8.388298394235687185927044901976, −8.133904360065490547704850284643, −8.023085879518547200053546288909, −6.82360311196668876877894694327, −6.73123274162529809512602491515, −6.23038313253906044283388714712, −5.42310621921974058590418778721, −5.24072665617237382596919834034, −5.21130807764064016373589120291, −4.36668333417167182683173318181, −3.85922393196806589048674666397, −3.61398722761819023593955849053, −2.78325880961878464635054640385, −2.59625636418181416770010524703, −2.13238907887844107604283289830, −1.28622004738090792605783465440, −0.941983646694215190241749713311, −0.30909090458710210277721397118, 0.30909090458710210277721397118, 0.941983646694215190241749713311, 1.28622004738090792605783465440, 2.13238907887844107604283289830, 2.59625636418181416770010524703, 2.78325880961878464635054640385, 3.61398722761819023593955849053, 3.85922393196806589048674666397, 4.36668333417167182683173318181, 5.21130807764064016373589120291, 5.24072665617237382596919834034, 5.42310621921974058590418778721, 6.23038313253906044283388714712, 6.73123274162529809512602491515, 6.82360311196668876877894694327, 8.023085879518547200053546288909, 8.133904360065490547704850284643, 8.388298394235687185927044901976, 9.744398851690347407339989222572, 10.26255468488312242667689075613, 10.67774157048055848572592132156, 11.25768378476704273024548085515, 11.62117347244427205983498729917, 11.90955870793936266870395636240, 12.39139236754961950827826358073