Dirichlet series
| L(s) = 1 | + 1.39e5·2-s − 1.04e8·3-s − 2.43e10·4-s + 8.92e11·5-s − 1.46e13·6-s + 8.78e14·7-s + 1.53e15·8-s + 5.49e15·9-s + 1.24e17·10-s − 1.15e18·11-s + 2.55e18·12-s − 6.21e19·13-s + 1.22e20·14-s − 9.36e19·15-s + 1.32e21·16-s − 3.93e21·17-s + 7.67e20·18-s − 3.23e22·19-s − 2.17e22·20-s − 9.21e22·21-s − 1.61e23·22-s − 5.14e23·23-s − 1.60e23·24-s − 3.64e24·25-s − 8.67e24·26-s + 1.06e25·27-s − 2.14e25·28-s + ⋯ |
| L(s) = 1 | + 0.753·2-s − 0.468·3-s − 0.709·4-s + 0.523·5-s − 0.353·6-s + 1.42·7-s + 0.240·8-s + 0.109·9-s + 0.394·10-s − 0.690·11-s + 0.332·12-s − 1.99·13-s + 1.07·14-s − 0.245·15-s + 1.11·16-s − 1.15·17-s + 0.0827·18-s − 1.35·19-s − 0.371·20-s − 0.669·21-s − 0.520·22-s − 0.761·23-s − 0.112·24-s − 1.25·25-s − 1.50·26-s + 0.954·27-s − 1.01·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(36-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+35/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(467.200\) |
| Root analytic conductor: | \(2.78559\) |
| Motivic weight: | \(35\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 1,\ (\ :35/2, 35/2, 35/2),\ 1)\) |
Particular Values
| \(L(18)\) | \(\approx\) | \(1.428702308\) |
| \(L(\frac12)\) | \(\approx\) | \(1.428702308\) |
| \(L(\frac{37}{2})\) | 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 | $S_4\times C_2$ | \( 1 - 17457 p^{3} T + 10710663 p^{12} T^{2} - 5276065965 p^{21} T^{3} + 10710663 p^{47} T^{4} - 17457 p^{73} T^{5} + p^{105} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 11652812 p^{2} T + 838400140897 p^{8} T^{2} - 248129113043430520 p^{16} T^{3} + 838400140897 p^{43} T^{4} + 11652812 p^{72} T^{5} + p^{105} T^{6} \) | |
| 5 | $S_4\times C_2$ | \( 1 - 178530410802 p T + \)\(14\!\cdots\!83\)\( p^{5} T^{2} - \)\(16\!\cdots\!04\)\( p^{10} T^{3} + \)\(14\!\cdots\!83\)\( p^{40} T^{4} - 178530410802 p^{71} T^{5} + p^{105} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 - 125488878478008 p T + \)\(36\!\cdots\!51\)\( p^{3} T^{2} - \)\(54\!\cdots\!00\)\( p^{6} T^{3} + \)\(36\!\cdots\!51\)\( p^{38} T^{4} - 125488878478008 p^{71} T^{5} + p^{105} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + 105267766231817004 p T + \)\(49\!\cdots\!15\)\( p^{3} T^{2} + \)\(31\!\cdots\!80\)\( p^{5} T^{3} + \)\(49\!\cdots\!15\)\( p^{38} T^{4} + 105267766231817004 p^{71} T^{5} + p^{105} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + 62139610550998650558 T + \)\(27\!\cdots\!79\)\( p T^{2} + \)\(55\!\cdots\!80\)\( p^{3} T^{3} + \)\(27\!\cdots\!79\)\( p^{36} T^{4} + 62139610550998650558 p^{70} T^{5} + p^{105} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(23\!\cdots\!82\)\( p T + \)\(11\!\cdots\!47\)\( p^{2} T^{2} + \)\(17\!\cdots\!20\)\( p^{3} T^{3} + \)\(11\!\cdots\!47\)\( p^{37} T^{4} + \)\(23\!\cdots\!82\)\( p^{71} T^{5} + p^{105} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(32\!\cdots\!40\)\( T + \)\(83\!\cdots\!63\)\( p T^{2} + \)\(10\!\cdots\!20\)\( p^{2} T^{3} + \)\(83\!\cdots\!63\)\( p^{36} T^{4} + \)\(32\!\cdots\!40\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 + \)\(22\!\cdots\!96\)\( p T + \)\(15\!\cdots\!53\)\( p^{2} T^{2} + \)\(21\!\cdots\!20\)\( p^{3} T^{3} + \)\(15\!\cdots\!53\)\( p^{37} T^{4} + \)\(22\!\cdots\!96\)\( p^{71} T^{5} + p^{105} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!90\)\( p T + \)\(36\!\cdots\!67\)\( p^{2} T^{2} + \)\(37\!\cdots\!20\)\( p^{3} T^{3} + \)\(36\!\cdots\!67\)\( p^{37} T^{4} + \)\(13\!\cdots\!90\)\( p^{71} T^{5} + p^{105} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(33\!\cdots\!76\)\( p T + \)\(48\!\cdots\!65\)\( p^{2} T^{2} - \)\(10\!\cdots\!20\)\( p^{3} T^{3} + \)\(48\!\cdots\!65\)\( p^{37} T^{4} - \)\(33\!\cdots\!76\)\( p^{71} T^{5} + p^{105} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 - \)\(24\!\cdots\!06\)\( T + \)\(58\!\cdots\!63\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(58\!\cdots\!63\)\( p^{35} T^{4} - \)\(24\!\cdots\!06\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(57\!\cdots\!66\)\( p T + \)\(83\!\cdots\!15\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(83\!\cdots\!15\)\( p^{35} T^{4} - \)\(57\!\cdots\!66\)\( p^{71} T^{5} + p^{105} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(47\!\cdots\!08\)\( T + \)\(45\!\cdots\!57\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!57\)\( p^{35} T^{4} + \)\(47\!\cdots\!08\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 - \)\(16\!\cdots\!56\)\( T + \)\(69\!\cdots\!53\)\( T^{2} - \)\(62\!\cdots\!60\)\( T^{3} + \)\(69\!\cdots\!53\)\( p^{35} T^{4} - \)\(16\!\cdots\!56\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(16\!\cdots\!58\)\( T + \)\(28\!\cdots\!67\)\( T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(28\!\cdots\!67\)\( p^{35} T^{4} + \)\(16\!\cdots\!58\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(43\!\cdots\!80\)\( T + \)\(26\!\cdots\!97\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!97\)\( p^{35} T^{4} - \)\(43\!\cdots\!80\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 - \)\(23\!\cdots\!06\)\( T + \)\(92\!\cdots\!15\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(92\!\cdots\!15\)\( p^{35} T^{4} - \)\(23\!\cdots\!06\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!44\)\( T + \)\(27\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!60\)\( T^{3} + \)\(27\!\cdots\!33\)\( p^{35} T^{4} + \)\(18\!\cdots\!44\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 - \)\(34\!\cdots\!56\)\( T + \)\(16\!\cdots\!65\)\( T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!65\)\( p^{35} T^{4} - \)\(34\!\cdots\!56\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(28\!\cdots\!58\)\( T + \)\(22\!\cdots\!87\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!87\)\( p^{35} T^{4} + \)\(28\!\cdots\!58\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(42\!\cdots\!60\)\( T + \)\(17\!\cdots\!97\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!97\)\( p^{35} T^{4} + \)\(42\!\cdots\!60\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(14\!\cdots\!92\)\( T + \)\(11\!\cdots\!97\)\( T^{2} - \)\(54\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!97\)\( p^{35} T^{4} - \)\(14\!\cdots\!92\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(30\!\cdots\!70\)\( T + \)\(80\!\cdots\!47\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(80\!\cdots\!47\)\( p^{35} T^{4} - \)\(30\!\cdots\!70\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(10\!\cdots\!94\)\( T + \)\(12\!\cdots\!03\)\( T^{2} + \)\(72\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!03\)\( p^{35} T^{4} + \)\(10\!\cdots\!94\)\( p^{70} T^{5} + p^{105} T^{6} \) | |
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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
−21.60990521007096499237175467177, −20.90277113678176266986213773335, −19.84115990496816626692568244831, −19.22014790230308983267427916080, −18.00626150863511425985242461230, −17.59830144349165301023594214509, −17.16519091384425771067062733987, −16.11351508686302067000630521883, −14.67016831171737410675322590588, −14.63850500192579912510421319527, −13.54187465949206101269929955780, −13.05924299969805232161932275817, −12.14124545628775680868267838282, −11.10643246241256940705304121290, −10.34826311131646356683386295159, −9.459703826379898497888748758919, −8.157521796120929573693857359706, −7.56742165052681597633141970493, −6.18932512378858861994327713434, −4.94822276660157313077474188750, −4.84357168970594102502750076522, −4.13950803345000955418253124047, −2.28810375041348958159043620596, −1.83589775632061795320516641947, −0.37295365828732270267756485604, 0.37295365828732270267756485604, 1.83589775632061795320516641947, 2.28810375041348958159043620596, 4.13950803345000955418253124047, 4.84357168970594102502750076522, 4.94822276660157313077474188750, 6.18932512378858861994327713434, 7.56742165052681597633141970493, 8.157521796120929573693857359706, 9.459703826379898497888748758919, 10.34826311131646356683386295159, 11.10643246241256940705304121290, 12.14124545628775680868267838282, 13.05924299969805232161932275817, 13.54187465949206101269929955780, 14.63850500192579912510421319527, 14.67016831171737410675322590588, 16.11351508686302067000630521883, 17.16519091384425771067062733987, 17.59830144349165301023594214509, 18.00626150863511425985242461230, 19.22014790230308983267427916080, 19.84115990496816626692568244831, 20.90277113678176266986213773335, 21.60990521007096499237175467177