Dirichlet series
| L(s) = 1 | − 4.12e11·2-s + 1.24e18·3-s + 1.13e23·4-s + 1.64e26·5-s − 5.14e29·6-s + 4.91e31·7-s − 2.59e34·8-s + 4.04e34·9-s − 6.79e37·10-s + 3.76e38·11-s + 1.41e41·12-s − 9.25e41·13-s − 2.02e43·14-s + 2.05e44·15-s + 5.35e45·16-s + 1.11e46·17-s − 1.66e46·18-s + 1.22e48·19-s + 1.86e49·20-s + 6.13e49·21-s − 1.55e50·22-s + 2.71e50·23-s − 3.23e52·24-s − 1.02e52·25-s + 3.81e53·26-s − 4.50e53·27-s + 5.57e54·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.59·3-s + 3·4-s + 1.01·5-s − 3.39·6-s + 1.00·7-s − 3.53·8-s + 0.0665·9-s − 2.14·10-s + 0.333·11-s + 4.79·12-s − 1.56·13-s − 2.12·14-s + 1.62·15-s + 15/4·16-s + 0.801·17-s − 0.141·18-s + 1.36·19-s + 3.03·20-s + 1.60·21-s − 0.707·22-s + 0.233·23-s − 5.65·24-s − 0.387·25-s + 3.31·26-s − 0.948·27-s + 3.00·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+75/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(361639.\) |
| Root analytic conductor: | \(8.44071\) |
| Motivic weight: | \(75\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 8,\ (\ :75/2, 75/2, 75/2),\ 1)\) |
Particular Values
| \(L(38)\) | \(\approx\) | \(3.937481465\) |
| \(L(\frac12)\) | \(\approx\) | \(3.937481465\) |
| \(L(\frac{77}{2})\) | 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 | $C_1$ | \( ( 1 + p^{37} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 415752245544732668 p T + \)\(25\!\cdots\!57\)\( p^{10} T^{2} - \)\(13\!\cdots\!96\)\( p^{21} T^{3} + \)\(25\!\cdots\!57\)\( p^{85} T^{4} - 415752245544732668 p^{151} T^{5} + p^{225} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - \)\(26\!\cdots\!82\)\( p^{4} T + \)\(76\!\cdots\!51\)\( p^{11} T^{2} - \)\(10\!\cdots\!32\)\( p^{21} T^{3} + \)\(76\!\cdots\!51\)\( p^{86} T^{4} - \)\(26\!\cdots\!82\)\( p^{154} T^{5} + p^{225} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 - \)\(70\!\cdots\!16\)\( p T + \)\(37\!\cdots\!73\)\( p^{6} T^{2} - \)\(16\!\cdots\!28\)\( p^{13} T^{3} + \)\(37\!\cdots\!73\)\( p^{81} T^{4} - \)\(70\!\cdots\!16\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 - \)\(34\!\cdots\!36\)\( p T + \)\(20\!\cdots\!75\)\( p^{3} T^{2} - \)\(72\!\cdots\!60\)\( p^{6} T^{3} + \)\(20\!\cdots\!75\)\( p^{78} T^{4} - \)\(34\!\cdots\!36\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + \)\(71\!\cdots\!82\)\( p T + \)\(29\!\cdots\!43\)\( p^{4} T^{2} + \)\(43\!\cdots\!64\)\( p^{9} T^{3} + \)\(29\!\cdots\!43\)\( p^{79} T^{4} + \)\(71\!\cdots\!82\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(65\!\cdots\!86\)\( p T + \)\(35\!\cdots\!79\)\( p^{3} T^{2} - \)\(43\!\cdots\!52\)\( p^{7} T^{3} + \)\(35\!\cdots\!79\)\( p^{78} T^{4} - \)\(65\!\cdots\!86\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 - \)\(64\!\cdots\!00\)\( p T + \)\(44\!\cdots\!77\)\( p^{2} T^{2} - \)\(30\!\cdots\!00\)\( p^{3} T^{3} + \)\(44\!\cdots\!77\)\( p^{77} T^{4} - \)\(64\!\cdots\!00\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(11\!\cdots\!28\)\( p T + \)\(39\!\cdots\!77\)\( p^{2} T^{2} + \)\(60\!\cdots\!44\)\( p^{5} T^{3} + \)\(39\!\cdots\!77\)\( p^{77} T^{4} - \)\(11\!\cdots\!28\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 + \)\(21\!\cdots\!70\)\( T + \)\(15\!\cdots\!67\)\( p^{2} T^{2} + \)\(30\!\cdots\!60\)\( p^{4} T^{3} + \)\(15\!\cdots\!67\)\( p^{77} T^{4} + \)\(21\!\cdots\!70\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 + \)\(84\!\cdots\!64\)\( T + \)\(51\!\cdots\!35\)\( p T^{2} + \)\(38\!\cdots\!00\)\( p^{3} T^{3} + \)\(51\!\cdots\!35\)\( p^{76} T^{4} + \)\(84\!\cdots\!64\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(17\!\cdots\!14\)\( p T + \)\(65\!\cdots\!23\)\( p^{2} T^{2} + \)\(98\!\cdots\!88\)\( p^{3} T^{3} + \)\(65\!\cdots\!23\)\( p^{77} T^{4} + \)\(17\!\cdots\!14\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(49\!\cdots\!74\)\( T + \)\(63\!\cdots\!95\)\( p T^{2} + \)\(48\!\cdots\!60\)\( p^{2} T^{3} + \)\(63\!\cdots\!95\)\( p^{76} T^{4} + \)\(49\!\cdots\!74\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(20\!\cdots\!96\)\( T + \)\(10\!\cdots\!93\)\( T^{2} + \)\(31\!\cdots\!84\)\( p T^{3} + \)\(10\!\cdots\!93\)\( p^{75} T^{4} + \)\(20\!\cdots\!96\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 - \)\(51\!\cdots\!52\)\( T + \)\(11\!\cdots\!51\)\( p T^{2} - \)\(13\!\cdots\!64\)\( p^{2} T^{3} + \)\(11\!\cdots\!51\)\( p^{76} T^{4} - \)\(51\!\cdots\!52\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(12\!\cdots\!38\)\( p T + \)\(14\!\cdots\!67\)\( p^{2} T^{2} - \)\(87\!\cdots\!84\)\( p^{3} T^{3} + \)\(14\!\cdots\!67\)\( p^{77} T^{4} - \)\(12\!\cdots\!38\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(51\!\cdots\!60\)\( T + \)\(43\!\cdots\!83\)\( p T^{2} - \)\(19\!\cdots\!80\)\( p^{2} T^{3} + \)\(43\!\cdots\!83\)\( p^{76} T^{4} - \)\(51\!\cdots\!60\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 - \)\(10\!\cdots\!06\)\( p T + \)\(58\!\cdots\!55\)\( p^{2} T^{2} - \)\(37\!\cdots\!80\)\( p^{3} T^{3} + \)\(58\!\cdots\!55\)\( p^{77} T^{4} - \)\(10\!\cdots\!06\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!68\)\( T + \)\(37\!\cdots\!11\)\( p T^{2} + \)\(55\!\cdots\!76\)\( p^{2} T^{3} + \)\(37\!\cdots\!11\)\( p^{76} T^{4} + \)\(13\!\cdots\!68\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 - \)\(14\!\cdots\!96\)\( p T + \)\(10\!\cdots\!05\)\( p^{2} T^{2} - \)\(50\!\cdots\!80\)\( p^{3} T^{3} + \)\(10\!\cdots\!05\)\( p^{77} T^{4} - \)\(14\!\cdots\!96\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 - \)\(32\!\cdots\!98\)\( p T + \)\(57\!\cdots\!67\)\( p^{2} T^{2} - \)\(68\!\cdots\!64\)\( p^{3} T^{3} + \)\(57\!\cdots\!67\)\( p^{77} T^{4} - \)\(32\!\cdots\!98\)\( p^{151} T^{5} + p^{225} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(20\!\cdots\!20\)\( T + \)\(67\!\cdots\!43\)\( p T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(67\!\cdots\!43\)\( p^{76} T^{4} - \)\(20\!\cdots\!20\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(11\!\cdots\!04\)\( T + \)\(11\!\cdots\!93\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(11\!\cdots\!93\)\( p^{75} T^{4} - \)\(11\!\cdots\!04\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(39\!\cdots\!70\)\( T + \)\(94\!\cdots\!47\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(94\!\cdots\!47\)\( p^{75} T^{4} + \)\(39\!\cdots\!70\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(39\!\cdots\!78\)\( T - \)\(34\!\cdots\!93\)\( T^{2} - \)\(63\!\cdots\!16\)\( T^{3} - \)\(34\!\cdots\!93\)\( p^{75} T^{4} + \)\(39\!\cdots\!78\)\( p^{150} T^{5} + p^{225} T^{6} \) | |
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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
−11.69477496203576396784150557796, −11.33573422479990395386948503628, −10.74931622088717196271561565925, −10.08535854157131868080875053716, −9.615820511184229326661561775550, −9.567747672167328515128594827500, −9.172479829280682284372343035216, −8.288330783217853007326929080582, −8.214874778419337713668843499059, −8.149838076635517335333226031925, −7.14142206508146716606849447384, −7.13758373683928759329064186547, −6.47805313487667273496264150563, −5.61338238573709102569140064268, −5.17664645352531999028967262865, −5.05526388598462771098677011113, −3.57235284911260475532528022149, −3.43532005742058874247842214448, −2.99491477356217662861848751487, −2.20741526402792460655730107632, −2.02843695096353448089937647280, −2.00408037055172795433389710703, −1.28989797488801820929982837747, −0.67672946533634429465130099953, −0.44360186992874782788816496510, 0.44360186992874782788816496510, 0.67672946533634429465130099953, 1.28989797488801820929982837747, 2.00408037055172795433389710703, 2.02843695096353448089937647280, 2.20741526402792460655730107632, 2.99491477356217662861848751487, 3.43532005742058874247842214448, 3.57235284911260475532528022149, 5.05526388598462771098677011113, 5.17664645352531999028967262865, 5.61338238573709102569140064268, 6.47805313487667273496264150563, 7.13758373683928759329064186547, 7.14142206508146716606849447384, 8.149838076635517335333226031925, 8.214874778419337713668843499059, 8.288330783217853007326929080582, 9.172479829280682284372343035216, 9.567747672167328515128594827500, 9.615820511184229326661561775550, 10.08535854157131868080875053716, 10.74931622088717196271561565925, 11.33573422479990395386948503628, 11.69477496203576396784150557796