Dirichlet series
| L(s) = 1 | − 2.42e7·2-s − 3.26e11·3-s − 5.35e14·4-s + 6.38e16·5-s + 7.92e18·6-s + 5.09e17·7-s + 2.65e22·8-s − 1.34e23·9-s − 1.54e24·10-s − 2.08e25·11-s + 1.75e26·12-s − 1.87e26·13-s − 1.23e25·14-s − 2.08e28·15-s − 3.42e29·16-s − 3.30e30·17-s + 3.26e30·18-s − 4.58e31·19-s − 3.41e31·20-s − 1.66e29·21-s + 5.04e32·22-s + 4.53e33·23-s − 8.66e33·24-s − 3.15e34·25-s + 4.54e33·26-s − 2.37e34·27-s − 2.72e32·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 0.668·3-s − 0.950·4-s + 0.479·5-s + 0.682·6-s + 0.00100·7-s + 1.98·8-s − 0.562·9-s − 0.489·10-s − 0.637·11-s + 0.635·12-s − 0.0958·13-s − 0.00102·14-s − 0.320·15-s − 1.08·16-s − 2.36·17-s + 0.574·18-s − 2.14·19-s − 0.455·20-s − 0.000671·21-s + 0.651·22-s + 1.96·23-s − 1.32·24-s − 1.77·25-s + 0.0978·26-s − 0.202·27-s − 0.000955·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(50-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+49/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(1\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3516.39\) |
| Root analytic conductor: | \(3.89956\) |
| Motivic weight: | \(49\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 1,\ (\ :49/2, 49/2, 49/2),\ -1)\) |
Particular Values
| \(L(25)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{51}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| good | 2 | $S_4\times C_2$ | \( 1 + 1514073 p^{4} T + 273950910267 p^{12} T^{2} + 406510566643005 p^{25} T^{3} + 273950910267 p^{61} T^{4} + 1514073 p^{102} T^{5} + p^{147} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 4036477684 p^{4} T + 4091277312609581777 p^{10} T^{2} + \)\(37\!\cdots\!40\)\( p^{18} T^{3} + 4091277312609581777 p^{59} T^{4} + 4036477684 p^{102} T^{5} + p^{147} T^{6} \) | |
| 5 | $S_4\times C_2$ | \( 1 - 511072285736634 p^{3} T + \)\(45\!\cdots\!59\)\( p^{7} T^{2} - \)\(90\!\cdots\!36\)\( p^{12} T^{3} + \)\(45\!\cdots\!59\)\( p^{56} T^{4} - 511072285736634 p^{101} T^{5} + p^{147} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 - 212158049770392 p^{4} T + \)\(44\!\cdots\!93\)\( p^{6} T^{2} - \)\(10\!\cdots\!00\)\( p^{9} T^{3} + \)\(44\!\cdots\!93\)\( p^{55} T^{4} - 212158049770392 p^{102} T^{5} + p^{147} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!84\)\( p T + \)\(14\!\cdots\!15\)\( p^{3} T^{2} + \)\(23\!\cdots\!80\)\( p^{6} T^{3} + \)\(14\!\cdots\!15\)\( p^{52} T^{4} + \)\(18\!\cdots\!84\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + \)\(14\!\cdots\!78\)\( p T + \)\(34\!\cdots\!99\)\( p^{3} T^{2} + \)\(15\!\cdots\!80\)\( p^{6} T^{3} + \)\(34\!\cdots\!99\)\( p^{52} T^{4} + \)\(14\!\cdots\!78\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(33\!\cdots\!38\)\( T + \)\(38\!\cdots\!71\)\( p T^{2} + \)\(19\!\cdots\!60\)\( p^{3} T^{3} + \)\(38\!\cdots\!71\)\( p^{50} T^{4} + \)\(33\!\cdots\!38\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!40\)\( p T + \)\(48\!\cdots\!17\)\( p^{2} T^{2} + \)\(31\!\cdots\!80\)\( p^{4} T^{3} + \)\(48\!\cdots\!17\)\( p^{51} T^{4} + \)\(24\!\cdots\!40\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(19\!\cdots\!12\)\( p T + \)\(20\!\cdots\!77\)\( p^{2} T^{2} - \)\(65\!\cdots\!20\)\( p^{4} T^{3} + \)\(20\!\cdots\!77\)\( p^{51} T^{4} - \)\(19\!\cdots\!12\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(11\!\cdots\!10\)\( T + \)\(60\!\cdots\!83\)\( p T^{2} - \)\(12\!\cdots\!80\)\( p^{2} T^{3} + \)\(60\!\cdots\!83\)\( p^{50} T^{4} - \)\(11\!\cdots\!10\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(93\!\cdots\!16\)\( T + \)\(31\!\cdots\!15\)\( p T^{2} + \)\(31\!\cdots\!80\)\( p^{2} T^{3} + \)\(31\!\cdots\!15\)\( p^{50} T^{4} - \)\(93\!\cdots\!16\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 - \)\(65\!\cdots\!46\)\( p T + \)\(12\!\cdots\!03\)\( p^{2} T^{2} - \)\(60\!\cdots\!20\)\( p^{3} T^{3} + \)\(12\!\cdots\!03\)\( p^{51} T^{4} - \)\(65\!\cdots\!46\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!54\)\( p T + \)\(24\!\cdots\!15\)\( p^{2} T^{2} + \)\(19\!\cdots\!80\)\( p^{3} T^{3} + \)\(24\!\cdots\!15\)\( p^{51} T^{4} + \)\(15\!\cdots\!54\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!08\)\( p T + \)\(15\!\cdots\!57\)\( p^{2} T^{2} + \)\(20\!\cdots\!00\)\( p^{3} T^{3} + \)\(15\!\cdots\!57\)\( p^{51} T^{4} + \)\(18\!\cdots\!08\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!76\)\( p T + \)\(11\!\cdots\!73\)\( p^{2} T^{2} - \)\(10\!\cdots\!60\)\( p^{3} T^{3} + \)\(11\!\cdots\!73\)\( p^{51} T^{4} - \)\(15\!\cdots\!76\)\( p^{99} T^{5} + p^{147} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!54\)\( T + \)\(82\!\cdots\!23\)\( T^{2} + \)\(50\!\cdots\!60\)\( T^{3} + \)\(82\!\cdots\!23\)\( p^{49} T^{4} + \)\(15\!\cdots\!54\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 + \)\(62\!\cdots\!80\)\( T + \)\(28\!\cdots\!17\)\( T^{2} + \)\(75\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!17\)\( p^{49} T^{4} + \)\(62\!\cdots\!80\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!74\)\( T + \)\(61\!\cdots\!15\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!15\)\( p^{49} T^{4} + \)\(24\!\cdots\!74\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(10\!\cdots\!88\)\( T + \)\(61\!\cdots\!57\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!57\)\( p^{49} T^{4} + \)\(10\!\cdots\!88\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(32\!\cdots\!04\)\( T + \)\(17\!\cdots\!65\)\( T^{2} + \)\(32\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!65\)\( p^{49} T^{4} + \)\(32\!\cdots\!04\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 - \)\(56\!\cdots\!26\)\( T + \)\(30\!\cdots\!83\)\( T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!83\)\( p^{49} T^{4} - \)\(56\!\cdots\!26\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(78\!\cdots\!40\)\( T + \)\(49\!\cdots\!57\)\( T^{2} + \)\(50\!\cdots\!20\)\( T^{3} + \)\(49\!\cdots\!57\)\( p^{49} T^{4} + \)\(78\!\cdots\!40\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(94\!\cdots\!16\)\( T + \)\(14\!\cdots\!13\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!13\)\( p^{49} T^{4} - \)\(94\!\cdots\!16\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(36\!\cdots\!30\)\( T + \)\(37\!\cdots\!27\)\( T^{2} - \)\(88\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!27\)\( p^{49} T^{4} - \)\(36\!\cdots\!30\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(10\!\cdots\!78\)\( T + \)\(73\!\cdots\!07\)\( T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(73\!\cdots\!07\)\( p^{49} T^{4} + \)\(10\!\cdots\!78\)\( p^{98} T^{5} + p^{147} T^{6} \) | |
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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
−18.66258266906742905187920213838, −17.90380556562652169931080960624, −17.42533108564586364924787074572, −17.36619134628449050722607351470, −16.59357600966871094712988073268, −15.49229808217256898622175699422, −14.99119836444912637958177980122, −13.64577534776753551998143639650, −13.48145910801877039197681375547, −12.97200104867324834434759329215, −11.75701241907280302598158799096, −10.78289761962258789745604528082, −10.72193136381819341197236094422, −9.584881447441338852328025424151, −9.099084253729332542690739709602, −8.511098314252081387804742690107, −8.008874204200169355909432833896, −6.68829227134675823405056170950, −6.33339507006627846871768884874, −5.19645735666897088606660426356, −4.58684981886443562953941294497, −4.27851438882376458693410009900, −2.82882696054372990146097340768, −2.06803731890084326163634144588, −1.36617840071946262905158063342, 0, 0, 0, 1.36617840071946262905158063342, 2.06803731890084326163634144588, 2.82882696054372990146097340768, 4.27851438882376458693410009900, 4.58684981886443562953941294497, 5.19645735666897088606660426356, 6.33339507006627846871768884874, 6.68829227134675823405056170950, 8.008874204200169355909432833896, 8.511098314252081387804742690107, 9.099084253729332542690739709602, 9.584881447441338852328025424151, 10.72193136381819341197236094422, 10.78289761962258789745604528082, 11.75701241907280302598158799096, 12.97200104867324834434759329215, 13.48145910801877039197681375547, 13.64577534776753551998143639650, 14.99119836444912637958177980122, 15.49229808217256898622175699422, 16.59357600966871094712988073268, 17.36619134628449050722607351470, 17.42533108564586364924787074572, 17.90380556562652169931080960624, 18.66258266906742905187920213838