Dirichlet series
| L(s) = 1 | + 2.63e13·2-s − 3.22e20·3-s + 4.64e26·4-s + 1.11e30·5-s − 8.50e33·6-s − 2.89e36·7-s + 6.80e39·8-s − 4.03e41·9-s + 2.94e43·10-s − 1.92e45·11-s − 1.49e47·12-s − 9.04e47·13-s − 7.64e49·14-s − 3.60e50·15-s + 8.97e52·16-s − 2.09e53·17-s − 1.06e55·18-s − 9.12e54·19-s + 5.18e56·20-s + 9.33e56·21-s − 5.07e58·22-s + 1.87e58·23-s − 2.19e60·24-s − 1.41e61·25-s − 2.38e61·26-s + 8.42e61·27-s − 1.34e63·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.566·3-s + 3·4-s + 0.439·5-s − 1.20·6-s − 0.501·7-s + 3.53·8-s − 1.24·9-s + 0.932·10-s − 0.963·11-s − 1.70·12-s − 0.316·13-s − 1.06·14-s − 0.249·15-s + 15/4·16-s − 0.626·17-s − 2.64·18-s − 0.216·19-s + 1.31·20-s + 0.284·21-s − 2.04·22-s + 0.109·23-s − 2.00·24-s − 2.18·25-s − 0.670·26-s + 0.458·27-s − 1.50·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(88-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+87/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(881056.\) |
| Root analytic conductor: | \(9.79115\) |
| Motivic weight: | \(87\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 8,\ (\ :87/2, 87/2, 87/2),\ -1)\) |
Particular Values
| \(L(44)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{89}{2})\) | 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 | $C_1$ | \( ( 1 - p^{43} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 1326186171032449012 p^{5} T + \)\(35\!\cdots\!59\)\( p^{15} T^{2} + \)\(33\!\cdots\!56\)\( p^{31} T^{3} + \)\(35\!\cdots\!59\)\( p^{102} T^{4} + 1326186171032449012 p^{179} T^{5} + p^{261} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - \)\(89\!\cdots\!66\)\( p^{3} T + \)\(31\!\cdots\!39\)\( p^{11} T^{2} - \)\(61\!\cdots\!92\)\( p^{22} T^{3} + \)\(31\!\cdots\!39\)\( p^{98} T^{4} - \)\(89\!\cdots\!66\)\( p^{177} T^{5} + p^{261} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + \)\(41\!\cdots\!84\)\( p T + \)\(55\!\cdots\!11\)\( p^{5} T^{2} + \)\(20\!\cdots\!72\)\( p^{13} T^{3} + \)\(55\!\cdots\!11\)\( p^{92} T^{4} + \)\(41\!\cdots\!84\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!64\)\( p^{2} T + \)\(90\!\cdots\!75\)\( p^{5} T^{2} - \)\(31\!\cdots\!60\)\( p^{9} T^{3} + \)\(90\!\cdots\!75\)\( p^{92} T^{4} + \)\(15\!\cdots\!64\)\( p^{176} T^{5} + p^{261} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + \)\(69\!\cdots\!62\)\( p T + \)\(76\!\cdots\!83\)\( p^{4} T^{2} + \)\(14\!\cdots\!44\)\( p^{9} T^{3} + \)\(76\!\cdots\!83\)\( p^{91} T^{4} + \)\(69\!\cdots\!62\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(72\!\cdots\!02\)\( p^{2} T + \)\(10\!\cdots\!71\)\( p^{5} T^{2} + \)\(80\!\cdots\!04\)\( p^{8} T^{3} + \)\(10\!\cdots\!71\)\( p^{92} T^{4} + \)\(72\!\cdots\!02\)\( p^{176} T^{5} + p^{261} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(48\!\cdots\!00\)\( p T + \)\(32\!\cdots\!63\)\( p^{3} T^{2} - \)\(46\!\cdots\!00\)\( p^{6} T^{3} + \)\(32\!\cdots\!63\)\( p^{90} T^{4} + \)\(48\!\cdots\!00\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(18\!\cdots\!84\)\( T + \)\(16\!\cdots\!91\)\( p T^{2} - \)\(32\!\cdots\!44\)\( p^{3} T^{3} + \)\(16\!\cdots\!91\)\( p^{88} T^{4} - \)\(18\!\cdots\!84\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(11\!\cdots\!70\)\( p T + \)\(50\!\cdots\!47\)\( p^{2} T^{2} - \)\(23\!\cdots\!60\)\( p^{3} T^{3} + \)\(50\!\cdots\!47\)\( p^{89} T^{4} - \)\(11\!\cdots\!70\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(83\!\cdots\!16\)\( T + \)\(57\!\cdots\!35\)\( p T^{2} - \)\(98\!\cdots\!00\)\( p^{2} T^{3} + \)\(57\!\cdots\!35\)\( p^{88} T^{4} - \)\(83\!\cdots\!16\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 - \)\(63\!\cdots\!26\)\( p T + \)\(45\!\cdots\!63\)\( p^{2} T^{2} - \)\(65\!\cdots\!96\)\( p^{4} T^{3} + \)\(45\!\cdots\!63\)\( p^{89} T^{4} - \)\(63\!\cdots\!26\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(29\!\cdots\!74\)\( p T + \)\(31\!\cdots\!95\)\( p^{3} T^{2} + \)\(50\!\cdots\!60\)\( p^{5} T^{3} + \)\(31\!\cdots\!95\)\( p^{90} T^{4} + \)\(29\!\cdots\!74\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(22\!\cdots\!72\)\( p T + \)\(12\!\cdots\!57\)\( p^{2} T^{2} + \)\(27\!\cdots\!16\)\( p^{3} T^{3} + \)\(12\!\cdots\!57\)\( p^{89} T^{4} + \)\(22\!\cdots\!72\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(50\!\cdots\!08\)\( T + \)\(49\!\cdots\!77\)\( T^{2} + \)\(45\!\cdots\!12\)\( p T^{3} + \)\(49\!\cdots\!77\)\( p^{87} T^{4} + \)\(50\!\cdots\!08\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(66\!\cdots\!06\)\( T + \)\(26\!\cdots\!91\)\( p T^{2} + \)\(60\!\cdots\!28\)\( p^{2} T^{3} + \)\(26\!\cdots\!91\)\( p^{88} T^{4} + \)\(66\!\cdots\!06\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(14\!\cdots\!60\)\( T + \)\(39\!\cdots\!23\)\( p T^{2} - \)\(10\!\cdots\!80\)\( p^{2} T^{3} + \)\(39\!\cdots\!23\)\( p^{88} T^{4} - \)\(14\!\cdots\!60\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 - \)\(54\!\cdots\!26\)\( T + \)\(86\!\cdots\!55\)\( p T^{2} - \)\(51\!\cdots\!80\)\( p^{2} T^{3} + \)\(86\!\cdots\!55\)\( p^{88} T^{4} - \)\(54\!\cdots\!26\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 - \)\(99\!\cdots\!92\)\( T + \)\(29\!\cdots\!71\)\( p T^{2} - \)\(32\!\cdots\!84\)\( p^{2} T^{3} + \)\(29\!\cdots\!71\)\( p^{88} T^{4} - \)\(99\!\cdots\!92\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(62\!\cdots\!64\)\( T + \)\(26\!\cdots\!55\)\( p T^{2} + \)\(72\!\cdots\!20\)\( p^{2} T^{3} + \)\(26\!\cdots\!55\)\( p^{88} T^{4} + \)\(62\!\cdots\!64\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(78\!\cdots\!22\)\( p T + \)\(36\!\cdots\!07\)\( p^{2} T^{2} + \)\(14\!\cdots\!16\)\( p^{3} T^{3} + \)\(36\!\cdots\!07\)\( p^{89} T^{4} + \)\(78\!\cdots\!22\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(26\!\cdots\!20\)\( T + \)\(47\!\cdots\!63\)\( p T^{2} - \)\(10\!\cdots\!60\)\( p^{2} T^{3} + \)\(47\!\cdots\!63\)\( p^{88} T^{4} - \)\(26\!\cdots\!20\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 + \)\(10\!\cdots\!72\)\( p T + \)\(62\!\cdots\!57\)\( p^{2} T^{2} - \)\(21\!\cdots\!84\)\( p^{3} T^{3} + \)\(62\!\cdots\!57\)\( p^{89} T^{4} + \)\(10\!\cdots\!72\)\( p^{175} T^{5} + p^{261} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(55\!\cdots\!30\)\( T + \)\(73\!\cdots\!87\)\( T^{2} - \)\(49\!\cdots\!40\)\( T^{3} + \)\(73\!\cdots\!87\)\( p^{87} T^{4} - \)\(55\!\cdots\!30\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(87\!\cdots\!38\)\( T + \)\(46\!\cdots\!87\)\( T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(46\!\cdots\!87\)\( p^{87} T^{4} + \)\(87\!\cdots\!38\)\( p^{174} T^{5} + p^{261} T^{6} \) | |
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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
−12.01483878130747954368909989026, −11.55537630380673684176511177223, −11.17418732768670391111284502323, −10.89421195603292099594427927019, −10.20433697774009455164584443916, −9.625110860310590459457723218420, −9.565641323378641633569488577006, −8.302041294875529015128684269177, −8.063930953941152305507304436573, −7.83181493691858884938318321540, −6.70692841807309708440926005677, −6.61102519709581192739482148140, −6.42321584058934397331075747569, −5.55282662178783676115181858669, −5.41926858206947779249228563708, −5.39265483838556770531654622632, −4.61990747071597836415756065717, −4.05267500699559602917734425279, −3.96721285049306411628655174086, −3.05890256000090809965131092828, −2.88947706489248499216351658144, −2.57994939793480002216417989503, −2.13391203631453267541258132350, −1.53114401983217974670993523428, −1.23345665677276392471285777637, 0, 0, 0, 1.23345665677276392471285777637, 1.53114401983217974670993523428, 2.13391203631453267541258132350, 2.57994939793480002216417989503, 2.88947706489248499216351658144, 3.05890256000090809965131092828, 3.96721285049306411628655174086, 4.05267500699559602917734425279, 4.61990747071597836415756065717, 5.39265483838556770531654622632, 5.41926858206947779249228563708, 5.55282662178783676115181858669, 6.42321584058934397331075747569, 6.61102519709581192739482148140, 6.70692841807309708440926005677, 7.83181493691858884938318321540, 8.063930953941152305507304436573, 8.302041294875529015128684269177, 9.565641323378641633569488577006, 9.625110860310590459457723218420, 10.20433697774009455164584443916, 10.89421195603292099594427927019, 11.17418732768670391111284502323, 11.55537630380673684176511177223, 12.01483878130747954368909989026