Dirichlet series
| L(s) = 1 | + 1.64e12·2-s − 4.58e18·3-s + 1.81e24·4-s − 4.05e27·5-s − 7.56e30·6-s + 1.48e33·7-s + 1.66e36·8-s − 6.67e37·9-s − 6.69e39·10-s − 1.24e41·11-s − 8.31e42·12-s + 4.27e43·13-s + 2.44e45·14-s + 1.86e46·15-s + 1.37e48·16-s + 9.46e47·17-s − 1.10e50·18-s + 2.02e49·19-s − 7.36e51·20-s − 6.80e51·21-s − 2.05e53·22-s + 8.25e52·23-s − 7.61e54·24-s − 6.03e54·25-s + 7.05e55·26-s + 5.05e56·27-s + 2.69e57·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.653·3-s + 3·4-s − 0.998·5-s − 1.38·6-s + 0.616·7-s + 3.53·8-s − 1.35·9-s − 2.11·10-s − 0.911·11-s − 1.95·12-s + 0.426·13-s + 1.30·14-s + 0.651·15-s + 15/4·16-s + 0.236·17-s − 2.87·18-s + 0.0624·19-s − 2.99·20-s − 0.402·21-s − 1.93·22-s + 0.134·23-s − 2.30·24-s − 0.364·25-s + 0.905·26-s + 1.46·27-s + 1.84·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(80-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+79/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(493927.\) |
| Root analytic conductor: | \(8.89086\) |
| Motivic weight: | \(79\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 8,\ (\ :79/2, 79/2, 79/2),\ -1)\) |
Particular Values
| \(L(40)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{81}{2})\) | 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 | $C_1$ | \( ( 1 - p^{39} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 169813679324026868 p^{3} T + \)\(18\!\cdots\!57\)\( p^{14} T^{2} + \)\(26\!\cdots\!96\)\( p^{27} T^{3} + \)\(18\!\cdots\!57\)\( p^{93} T^{4} + 169813679324026868 p^{161} T^{5} + p^{237} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + \)\(32\!\cdots\!46\)\( p^{3} T + \)\(46\!\cdots\!83\)\( p^{11} T^{2} + \)\(11\!\cdots\!72\)\( p^{21} T^{3} + \)\(46\!\cdots\!83\)\( p^{90} T^{4} + \)\(32\!\cdots\!46\)\( p^{161} T^{5} + p^{237} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 - \)\(21\!\cdots\!16\)\( p T + \)\(38\!\cdots\!77\)\( p^{4} T^{2} - \)\(73\!\cdots\!04\)\( p^{10} T^{3} + \)\(38\!\cdots\!77\)\( p^{83} T^{4} - \)\(21\!\cdots\!16\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!44\)\( p T + \)\(36\!\cdots\!75\)\( p^{5} T^{2} + \)\(19\!\cdots\!40\)\( p^{9} T^{3} + \)\(36\!\cdots\!75\)\( p^{84} T^{4} + \)\(11\!\cdots\!44\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(32\!\cdots\!58\)\( p T + \)\(82\!\cdots\!23\)\( p^{4} T^{2} - \)\(12\!\cdots\!44\)\( p^{7} T^{3} + \)\(82\!\cdots\!23\)\( p^{83} T^{4} - \)\(32\!\cdots\!58\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(55\!\cdots\!46\)\( p T + \)\(71\!\cdots\!27\)\( p^{4} T^{2} + \)\(72\!\cdots\!64\)\( p^{8} T^{3} + \)\(71\!\cdots\!27\)\( p^{83} T^{4} - \)\(55\!\cdots\!46\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 - \)\(10\!\cdots\!00\)\( p T + \)\(31\!\cdots\!43\)\( p^{3} T^{2} - \)\(34\!\cdots\!00\)\( p^{6} T^{3} + \)\(31\!\cdots\!43\)\( p^{82} T^{4} - \)\(10\!\cdots\!00\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(82\!\cdots\!24\)\( T + \)\(51\!\cdots\!59\)\( p^{3} T^{2} - \)\(67\!\cdots\!68\)\( p^{4} T^{3} + \)\(51\!\cdots\!59\)\( p^{82} T^{4} - \)\(82\!\cdots\!24\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(51\!\cdots\!70\)\( p T + \)\(62\!\cdots\!27\)\( p^{2} T^{2} - \)\(99\!\cdots\!40\)\( p^{4} T^{3} + \)\(62\!\cdots\!27\)\( p^{81} T^{4} - \)\(51\!\cdots\!70\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 + \)\(95\!\cdots\!04\)\( T + \)\(38\!\cdots\!35\)\( p T^{2} + \)\(69\!\cdots\!00\)\( p^{2} T^{3} + \)\(38\!\cdots\!35\)\( p^{80} T^{4} + \)\(95\!\cdots\!04\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(39\!\cdots\!34\)\( p T + \)\(17\!\cdots\!03\)\( p^{2} T^{2} + \)\(10\!\cdots\!84\)\( p^{4} T^{3} + \)\(17\!\cdots\!03\)\( p^{81} T^{4} + \)\(39\!\cdots\!34\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(89\!\cdots\!94\)\( T + \)\(16\!\cdots\!95\)\( p T^{2} + \)\(24\!\cdots\!60\)\( p^{2} T^{3} + \)\(16\!\cdots\!95\)\( p^{80} T^{4} + \)\(89\!\cdots\!94\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(95\!\cdots\!96\)\( T + \)\(43\!\cdots\!93\)\( T^{2} + \)\(33\!\cdots\!84\)\( p T^{3} + \)\(43\!\cdots\!93\)\( p^{79} T^{4} + \)\(95\!\cdots\!96\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!68\)\( T + \)\(14\!\cdots\!57\)\( T^{2} + \)\(55\!\cdots\!32\)\( p T^{3} + \)\(14\!\cdots\!57\)\( p^{79} T^{4} + \)\(11\!\cdots\!68\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(12\!\cdots\!74\)\( T + \)\(56\!\cdots\!31\)\( p T^{2} - \)\(16\!\cdots\!92\)\( p^{2} T^{3} + \)\(56\!\cdots\!31\)\( p^{80} T^{4} - \)\(12\!\cdots\!74\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!40\)\( T + \)\(34\!\cdots\!63\)\( p T^{2} + \)\(50\!\cdots\!20\)\( p^{2} T^{3} + \)\(34\!\cdots\!63\)\( p^{80} T^{4} + \)\(11\!\cdots\!40\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(64\!\cdots\!14\)\( T + \)\(45\!\cdots\!55\)\( p T^{2} + \)\(24\!\cdots\!20\)\( p^{2} T^{3} + \)\(45\!\cdots\!55\)\( p^{80} T^{4} + \)\(64\!\cdots\!14\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 - \)\(10\!\cdots\!52\)\( T + \)\(84\!\cdots\!31\)\( p T^{2} - \)\(86\!\cdots\!44\)\( p^{2} T^{3} + \)\(84\!\cdots\!31\)\( p^{80} T^{4} - \)\(10\!\cdots\!52\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(66\!\cdots\!55\)\( p T^{2} + \)\(69\!\cdots\!20\)\( p^{2} T^{3} + \)\(66\!\cdots\!55\)\( p^{80} T^{4} + \)\(11\!\cdots\!44\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(28\!\cdots\!42\)\( p T + \)\(57\!\cdots\!47\)\( p^{2} T^{2} + \)\(63\!\cdots\!96\)\( p^{3} T^{3} + \)\(57\!\cdots\!47\)\( p^{81} T^{4} + \)\(28\!\cdots\!42\)\( p^{159} T^{5} + p^{237} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(75\!\cdots\!80\)\( T + \)\(14\!\cdots\!57\)\( T^{2} + \)\(77\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!57\)\( p^{79} T^{4} + \)\(75\!\cdots\!80\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 + \)\(26\!\cdots\!56\)\( T + \)\(34\!\cdots\!53\)\( T^{2} + \)\(27\!\cdots\!72\)\( T^{3} + \)\(34\!\cdots\!53\)\( p^{79} T^{4} + \)\(26\!\cdots\!56\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(29\!\cdots\!70\)\( T + \)\(56\!\cdots\!27\)\( T^{2} + \)\(67\!\cdots\!60\)\( T^{3} + \)\(56\!\cdots\!27\)\( p^{79} T^{4} + \)\(29\!\cdots\!70\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(35\!\cdots\!98\)\( T + \)\(25\!\cdots\!67\)\( T^{2} + \)\(55\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!67\)\( p^{79} T^{4} + \)\(35\!\cdots\!98\)\( p^{158} T^{5} + p^{237} T^{6} \) | |
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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
−12.32955254044194265661517998932, −11.74397567722722779603721845452, −11.55094817692017061353646919826, −11.46194391901550885237107426326, −10.84287122125300077107059439218, −10.32750403528094666710094672249, −9.938694593383628598907820005713, −8.646664693565706599706396467910, −8.425256241073473674646002341057, −8.091807527277091461251136934869, −7.25231175287166635444336440548, −7.13812590924770087482111290880, −6.46033889647388943277533573820, −5.87407062387562113133687186334, −5.66077261642081261445923183424, −5.16222825899425961174652250122, −4.80861865193965941424379036238, −4.55467713696389609377004316598, −3.81949989704460253595283965963, −3.25748674149963422995554692464, −3.19847960954068489244977756821, −2.83226907544269284568698794055, −1.88186552337273229405267214159, −1.69949070065376437110594529731, −1.25419273458959932703460050980, 0, 0, 0, 1.25419273458959932703460050980, 1.69949070065376437110594529731, 1.88186552337273229405267214159, 2.83226907544269284568698794055, 3.19847960954068489244977756821, 3.25748674149963422995554692464, 3.81949989704460253595283965963, 4.55467713696389609377004316598, 4.80861865193965941424379036238, 5.16222825899425961174652250122, 5.66077261642081261445923183424, 5.87407062387562113133687186334, 6.46033889647388943277533573820, 7.13812590924770087482111290880, 7.25231175287166635444336440548, 8.091807527277091461251136934869, 8.425256241073473674646002341057, 8.646664693565706599706396467910, 9.938694593383628598907820005713, 10.32750403528094666710094672249, 10.84287122125300077107059439218, 11.46194391901550885237107426326, 11.55094817692017061353646919826, 11.74397567722722779603721845452, 12.32955254044194265661517998932