Dirichlet series
| L(s) = 1 | + 1.28e10·2-s + 2.98e15·3-s + 1.10e20·4-s + 3.91e21·5-s + 3.84e25·6-s + 4.29e27·7-s + 7.92e29·8-s − 6.74e30·9-s + 5.04e31·10-s − 1.96e32·11-s + 3.30e35·12-s + 5.94e35·13-s + 5.53e37·14-s + 1.16e37·15-s + 5.10e39·16-s − 4.54e39·17-s − 8.69e40·18-s − 7.67e40·19-s + 4.33e41·20-s + 1.28e43·21-s − 2.53e42·22-s − 2.46e44·23-s + 2.36e45·24-s − 3.96e45·25-s + 7.65e45·26-s − 4.45e45·27-s + 4.75e47·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.929·3-s + 3·4-s + 0.0752·5-s + 1.97·6-s + 1.46·7-s + 3.53·8-s − 0.654·9-s + 0.159·10-s − 0.0280·11-s + 2.78·12-s + 0.372·13-s + 3.11·14-s + 0.0699·15-s + 15/4·16-s − 0.465·17-s − 1.38·18-s − 0.211·19-s + 0.225·20-s + 1.36·21-s − 0.0594·22-s − 1.36·23-s + 3.28·24-s − 1.46·25-s + 0.789·26-s − 0.134·27-s + 4.40·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+65/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(153254.\) |
| Root analytic conductor: | \(7.31535\) |
| Motivic weight: | \(65\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 8,\ (\ :65/2, 65/2, 65/2),\ 1)\) |
Particular Values
| \(L(33)\) | \(\approx\) | \(40.65415966\) |
| \(L(\frac12)\) | \(\approx\) | \(40.65415966\) |
| \(L(\frac{67}{2})\) | 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 | $C_1$ | \( ( 1 - p^{32} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 36846402447052 p^{4} T + \)\(26\!\cdots\!73\)\( p^{10} T^{2} - \)\(22\!\cdots\!16\)\( p^{24} T^{3} + \)\(26\!\cdots\!73\)\( p^{75} T^{4} - 36846402447052 p^{134} T^{5} + p^{195} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!22\)\( p^{2} T + \)\(63\!\cdots\!99\)\( p^{4} T^{2} + \)\(12\!\cdots\!36\)\( p^{13} T^{3} + \)\(63\!\cdots\!99\)\( p^{69} T^{4} - \)\(15\!\cdots\!22\)\( p^{132} T^{5} + p^{195} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 - \)\(12\!\cdots\!48\)\( p^{3} T + \)\(40\!\cdots\!79\)\( p^{9} T^{2} - \)\(22\!\cdots\!24\)\( p^{17} T^{3} + \)\(40\!\cdots\!79\)\( p^{74} T^{4} - \)\(12\!\cdots\!48\)\( p^{133} T^{5} + p^{195} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(17\!\cdots\!44\)\( p T + \)\(68\!\cdots\!55\)\( p^{3} T^{2} - \)\(55\!\cdots\!80\)\( p^{6} T^{3} + \)\(68\!\cdots\!55\)\( p^{68} T^{4} + \)\(17\!\cdots\!44\)\( p^{131} T^{5} + p^{195} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(45\!\cdots\!34\)\( p T + \)\(19\!\cdots\!11\)\( p^{3} T^{2} - \)\(74\!\cdots\!84\)\( p^{6} T^{3} + \)\(19\!\cdots\!11\)\( p^{68} T^{4} - \)\(45\!\cdots\!34\)\( p^{131} T^{5} + p^{195} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(45\!\cdots\!46\)\( T + \)\(81\!\cdots\!79\)\( p T^{2} + \)\(25\!\cdots\!24\)\( p^{3} T^{3} + \)\(81\!\cdots\!79\)\( p^{66} T^{4} + \)\(45\!\cdots\!46\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(40\!\cdots\!20\)\( p T + \)\(46\!\cdots\!83\)\( p^{3} T^{2} + \)\(30\!\cdots\!40\)\( p^{6} T^{3} + \)\(46\!\cdots\!83\)\( p^{68} T^{4} + \)\(40\!\cdots\!20\)\( p^{131} T^{5} + p^{195} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!68\)\( T + \)\(71\!\cdots\!19\)\( p T^{2} + \)\(27\!\cdots\!92\)\( p^{3} T^{3} + \)\(71\!\cdots\!19\)\( p^{66} T^{4} + \)\(24\!\cdots\!68\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(53\!\cdots\!10\)\( T + \)\(76\!\cdots\!43\)\( p T^{2} - \)\(24\!\cdots\!20\)\( p^{3} T^{3} + \)\(76\!\cdots\!43\)\( p^{66} T^{4} - \)\(53\!\cdots\!10\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(19\!\cdots\!96\)\( p T + \)\(10\!\cdots\!95\)\( p^{3} T^{2} - \)\(34\!\cdots\!40\)\( p^{5} T^{3} + \)\(10\!\cdots\!95\)\( p^{68} T^{4} - \)\(19\!\cdots\!96\)\( p^{131} T^{5} + p^{195} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(16\!\cdots\!06\)\( T + \)\(61\!\cdots\!59\)\( p T^{2} + \)\(20\!\cdots\!68\)\( p^{2} T^{3} + \)\(61\!\cdots\!59\)\( p^{66} T^{4} + \)\(16\!\cdots\!06\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(36\!\cdots\!86\)\( T + \)\(20\!\cdots\!35\)\( T^{2} - \)\(11\!\cdots\!00\)\( p T^{3} + \)\(20\!\cdots\!35\)\( p^{65} T^{4} - \)\(36\!\cdots\!86\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 - \)\(19\!\cdots\!92\)\( T + \)\(11\!\cdots\!19\)\( p T^{2} - \)\(31\!\cdots\!44\)\( p^{2} T^{3} + \)\(11\!\cdots\!19\)\( p^{66} T^{4} - \)\(19\!\cdots\!92\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!56\)\( T + \)\(21\!\cdots\!39\)\( p T^{2} + \)\(36\!\cdots\!88\)\( p^{2} T^{3} + \)\(21\!\cdots\!39\)\( p^{66} T^{4} + \)\(15\!\cdots\!56\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(38\!\cdots\!82\)\( T + \)\(15\!\cdots\!79\)\( p T^{2} - \)\(39\!\cdots\!04\)\( p^{2} T^{3} + \)\(15\!\cdots\!79\)\( p^{66} T^{4} - \)\(38\!\cdots\!82\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(99\!\cdots\!20\)\( T + \)\(10\!\cdots\!83\)\( p T^{2} - \)\(77\!\cdots\!60\)\( p^{2} T^{3} + \)\(10\!\cdots\!83\)\( p^{66} T^{4} - \)\(99\!\cdots\!20\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 - \)\(50\!\cdots\!86\)\( p T + \)\(15\!\cdots\!75\)\( p^{2} T^{2} - \)\(29\!\cdots\!60\)\( p^{3} T^{3} + \)\(15\!\cdots\!75\)\( p^{67} T^{4} - \)\(50\!\cdots\!86\)\( p^{131} T^{5} + p^{195} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 - \)\(17\!\cdots\!64\)\( T + \)\(28\!\cdots\!53\)\( T^{2} + \)\(14\!\cdots\!32\)\( T^{3} + \)\(28\!\cdots\!53\)\( p^{65} T^{4} - \)\(17\!\cdots\!64\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(20\!\cdots\!04\)\( T + \)\(75\!\cdots\!25\)\( T^{2} + \)\(89\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!25\)\( p^{65} T^{4} + \)\(20\!\cdots\!04\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(14\!\cdots\!98\)\( T + \)\(31\!\cdots\!47\)\( T^{2} + \)\(25\!\cdots\!24\)\( T^{3} + \)\(31\!\cdots\!47\)\( p^{65} T^{4} + \)\(14\!\cdots\!98\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(61\!\cdots\!00\)\( T + \)\(52\!\cdots\!97\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!97\)\( p^{65} T^{4} - \)\(61\!\cdots\!00\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(20\!\cdots\!72\)\( T + \)\(84\!\cdots\!57\)\( T^{2} - \)\(58\!\cdots\!16\)\( T^{3} + \)\(84\!\cdots\!57\)\( p^{65} T^{4} - \)\(20\!\cdots\!72\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!30\)\( T + \)\(68\!\cdots\!47\)\( T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + \)\(68\!\cdots\!47\)\( p^{65} T^{4} + \)\(11\!\cdots\!30\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(74\!\cdots\!46\)\( T + \)\(59\!\cdots\!43\)\( T^{2} + \)\(21\!\cdots\!12\)\( T^{3} + \)\(59\!\cdots\!43\)\( p^{65} T^{4} + \)\(74\!\cdots\!46\)\( p^{130} T^{5} + p^{195} T^{6} \) | |
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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
−13.02262151801943087091315352467, −12.02302401125657699551334807788, −11.93530931973395509040108334330, −11.56907588211655378771318217010, −10.96650291763257909335616401582, −10.24708823574178167643630422131, −10.04212975486410247557076219410, −8.766633873035407594682105347598, −8.296716612835563553707527050017, −8.285150551462190852755256800285, −7.51824270436280006839472335201, −6.87424975320932289179146394437, −6.35506745690576753653094904462, −5.86303191000229961086496801721, −5.25665256081847827010448130546, −5.08626935017337662750911379832, −4.25521628636417732858790618043, −3.86007448495643775061002508817, −3.86007208291752441219529835717, −2.62427251309994093968122992386, −2.56587004394753263773104492944, −2.38542870140314331208010161254, −1.64727244612375915654148940377, −1.08431744292981397182448263762, −0.56747129256755184420359215793, 0.56747129256755184420359215793, 1.08431744292981397182448263762, 1.64727244612375915654148940377, 2.38542870140314331208010161254, 2.56587004394753263773104492944, 2.62427251309994093968122992386, 3.86007208291752441219529835717, 3.86007448495643775061002508817, 4.25521628636417732858790618043, 5.08626935017337662750911379832, 5.25665256081847827010448130546, 5.86303191000229961086496801721, 6.35506745690576753653094904462, 6.87424975320932289179146394437, 7.51824270436280006839472335201, 8.285150551462190852755256800285, 8.296716612835563553707527050017, 8.766633873035407594682105347598, 10.04212975486410247557076219410, 10.24708823574178167643630422131, 10.96650291763257909335616401582, 11.56907588211655378771318217010, 11.93530931973395509040108334330, 12.02302401125657699551334807788, 13.02262151801943087091315352467