Dirichlet series
| L(s) = 1 | + 6.61e10·2-s + 8.98e16·3-s − 7.87e20·4-s − 4.27e24·5-s + 5.94e27·6-s + 3.39e29·7-s − 1.13e32·8-s − 3.99e33·9-s − 2.83e35·10-s − 7.28e36·11-s − 7.07e37·12-s + 1.97e39·13-s + 2.24e40·14-s − 3.84e41·15-s − 2.88e42·16-s + 3.18e43·17-s − 2.64e44·18-s − 2.13e45·19-s + 3.36e45·20-s + 3.04e46·21-s − 4.81e47·22-s + 9.64e47·23-s − 1.01e49·24-s − 1.27e50·25-s + 1.30e50·26-s − 6.55e50·27-s − 2.67e50·28-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 1.03·3-s − 0.333·4-s − 0.657·5-s + 1.41·6-s + 0.338·7-s − 0.988·8-s − 0.531·9-s − 0.895·10-s − 0.781·11-s − 0.346·12-s + 0.564·13-s + 0.460·14-s − 0.681·15-s − 0.518·16-s + 0.663·17-s − 0.724·18-s − 0.858·19-s + 0.219·20-s + 0.351·21-s − 1.06·22-s + 0.439·23-s − 1.02·24-s − 3.01·25-s + 0.768·26-s − 1.00·27-s − 0.112·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(72-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+71/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.05865\times 10^{9}\) |
| Root analytic conductor: | \(5.65018\) |
| Motivic weight: | \(71\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 1,\ (\ :[71/2]^{6}),\ 1)\) |
Particular Values
| \(L(36)\) | \(\approx\) | \(0.0007417782330\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0007417782330\) |
| \(L(\frac{73}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| good | 2 | \( 1 - 8269667055 p^{3} T + 5043301488604424415 p^{10} T^{2} - \)\(13\!\cdots\!35\)\( p^{21} T^{3} + \)\(65\!\cdots\!73\)\( p^{38} T^{4} - \)\(68\!\cdots\!85\)\( p^{57} T^{5} + \)\(19\!\cdots\!95\)\( p^{78} T^{6} - \)\(68\!\cdots\!85\)\( p^{128} T^{7} + \)\(65\!\cdots\!73\)\( p^{180} T^{8} - \)\(13\!\cdots\!35\)\( p^{234} T^{9} + 5043301488604424415 p^{294} T^{10} - 8269667055 p^{358} T^{11} + p^{426} T^{12} \) |
| 3 | \( 1 - 9988550308641160 p^{2} T + \)\(61\!\cdots\!90\)\( p^{9} T^{2} - \)\(61\!\cdots\!60\)\( p^{17} T^{3} + \)\(37\!\cdots\!63\)\( p^{26} T^{4} - \)\(11\!\cdots\!40\)\( p^{41} T^{5} + \)\(15\!\cdots\!40\)\( p^{58} T^{6} - \)\(11\!\cdots\!40\)\( p^{112} T^{7} + \)\(37\!\cdots\!63\)\( p^{168} T^{8} - \)\(61\!\cdots\!60\)\( p^{230} T^{9} + \)\(61\!\cdots\!90\)\( p^{293} T^{10} - 9988550308641160 p^{357} T^{11} + p^{426} T^{12} \) | |
| 5 | \( 1 + \)\(85\!\cdots\!44\)\( p T + \)\(46\!\cdots\!26\)\( p^{5} T^{2} + \)\(66\!\cdots\!88\)\( p^{10} T^{3} + \)\(73\!\cdots\!99\)\( p^{16} T^{4} + \)\(28\!\cdots\!76\)\( p^{26} T^{5} + \)\(79\!\cdots\!04\)\( p^{37} T^{6} + \)\(28\!\cdots\!76\)\( p^{97} T^{7} + \)\(73\!\cdots\!99\)\( p^{158} T^{8} + \)\(66\!\cdots\!88\)\( p^{223} T^{9} + \)\(46\!\cdots\!26\)\( p^{289} T^{10} + \)\(85\!\cdots\!44\)\( p^{356} T^{11} + p^{426} T^{12} \) | |
| 7 | \( 1 - \)\(48\!\cdots\!00\)\( p T + \)\(48\!\cdots\!50\)\( p^{3} T^{2} - \)\(16\!\cdots\!00\)\( p^{7} T^{3} + \)\(20\!\cdots\!21\)\( p^{13} T^{4} - \)\(97\!\cdots\!00\)\( p^{20} T^{5} + \)\(54\!\cdots\!00\)\( p^{28} T^{6} - \)\(97\!\cdots\!00\)\( p^{91} T^{7} + \)\(20\!\cdots\!21\)\( p^{155} T^{8} - \)\(16\!\cdots\!00\)\( p^{220} T^{9} + \)\(48\!\cdots\!50\)\( p^{287} T^{10} - \)\(48\!\cdots\!00\)\( p^{356} T^{11} + p^{426} T^{12} \) | |
| 11 | \( 1 + \)\(72\!\cdots\!28\)\( T + \)\(14\!\cdots\!06\)\( p^{2} T^{2} - \)\(14\!\cdots\!20\)\( p^{5} T^{3} + \)\(54\!\cdots\!45\)\( p^{9} T^{4} - \)\(98\!\cdots\!12\)\( p^{14} T^{5} + \)\(27\!\cdots\!84\)\( p^{19} T^{6} - \)\(98\!\cdots\!12\)\( p^{85} T^{7} + \)\(54\!\cdots\!45\)\( p^{151} T^{8} - \)\(14\!\cdots\!20\)\( p^{218} T^{9} + \)\(14\!\cdots\!06\)\( p^{286} T^{10} + \)\(72\!\cdots\!28\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 13 | \( 1 - \)\(19\!\cdots\!80\)\( T + \)\(48\!\cdots\!30\)\( p T^{2} - \)\(48\!\cdots\!80\)\( p^{3} T^{3} + \)\(36\!\cdots\!23\)\( p^{6} T^{4} - \)\(17\!\cdots\!60\)\( p^{10} T^{5} + \)\(54\!\cdots\!60\)\( p^{15} T^{6} - \)\(17\!\cdots\!60\)\( p^{81} T^{7} + \)\(36\!\cdots\!23\)\( p^{148} T^{8} - \)\(48\!\cdots\!80\)\( p^{216} T^{9} + \)\(48\!\cdots\!30\)\( p^{285} T^{10} - \)\(19\!\cdots\!80\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 17 | \( 1 - \)\(18\!\cdots\!60\)\( p T + \)\(22\!\cdots\!70\)\( p^{2} T^{2} + \)\(20\!\cdots\!40\)\( p^{4} T^{3} + \)\(30\!\cdots\!79\)\( p^{7} T^{4} + \)\(39\!\cdots\!60\)\( p^{10} T^{5} + \)\(12\!\cdots\!20\)\( p^{13} T^{6} + \)\(39\!\cdots\!60\)\( p^{81} T^{7} + \)\(30\!\cdots\!79\)\( p^{149} T^{8} + \)\(20\!\cdots\!40\)\( p^{217} T^{9} + \)\(22\!\cdots\!70\)\( p^{286} T^{10} - \)\(18\!\cdots\!60\)\( p^{356} T^{11} + p^{426} T^{12} \) | |
| 19 | \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(11\!\cdots\!06\)\( p T^{2} + \)\(75\!\cdots\!00\)\( p^{2} T^{3} + \)\(18\!\cdots\!15\)\( p^{4} T^{4} + \)\(24\!\cdots\!00\)\( p^{7} T^{5} + \)\(29\!\cdots\!80\)\( p^{10} T^{6} + \)\(24\!\cdots\!00\)\( p^{78} T^{7} + \)\(18\!\cdots\!15\)\( p^{146} T^{8} + \)\(75\!\cdots\!00\)\( p^{215} T^{9} + \)\(11\!\cdots\!06\)\( p^{285} T^{10} + \)\(21\!\cdots\!80\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 23 | \( 1 - \)\(41\!\cdots\!40\)\( p T + \)\(36\!\cdots\!90\)\( p^{2} T^{2} - \)\(54\!\cdots\!40\)\( p^{4} T^{3} + \)\(12\!\cdots\!83\)\( p^{6} T^{4} - \)\(14\!\cdots\!60\)\( p^{8} T^{5} + \)\(25\!\cdots\!20\)\( p^{10} T^{6} - \)\(14\!\cdots\!60\)\( p^{79} T^{7} + \)\(12\!\cdots\!83\)\( p^{148} T^{8} - \)\(54\!\cdots\!40\)\( p^{217} T^{9} + \)\(36\!\cdots\!90\)\( p^{286} T^{10} - \)\(41\!\cdots\!40\)\( p^{356} T^{11} + p^{426} T^{12} \) | |
| 29 | \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(71\!\cdots\!06\)\( p T^{2} + \)\(21\!\cdots\!00\)\( p^{2} T^{3} + \)\(35\!\cdots\!15\)\( p^{4} T^{4} + \)\(12\!\cdots\!00\)\( p^{6} T^{5} + \)\(39\!\cdots\!80\)\( p^{8} T^{6} + \)\(12\!\cdots\!00\)\( p^{77} T^{7} + \)\(35\!\cdots\!15\)\( p^{146} T^{8} + \)\(21\!\cdots\!00\)\( p^{215} T^{9} + \)\(71\!\cdots\!06\)\( p^{285} T^{10} + \)\(13\!\cdots\!20\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 31 | \( 1 - \)\(20\!\cdots\!92\)\( T + \)\(17\!\cdots\!66\)\( p T^{2} - \)\(80\!\cdots\!20\)\( p^{2} T^{3} + \)\(12\!\cdots\!95\)\( p^{4} T^{4} - \)\(13\!\cdots\!32\)\( p^{6} T^{5} + \)\(14\!\cdots\!24\)\( p^{8} T^{6} - \)\(13\!\cdots\!32\)\( p^{77} T^{7} + \)\(12\!\cdots\!95\)\( p^{146} T^{8} - \)\(80\!\cdots\!20\)\( p^{215} T^{9} + \)\(17\!\cdots\!66\)\( p^{285} T^{10} - \)\(20\!\cdots\!92\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 37 | \( 1 - \)\(20\!\cdots\!60\)\( T + \)\(27\!\cdots\!90\)\( T^{2} - \)\(25\!\cdots\!80\)\( T^{3} + \)\(51\!\cdots\!11\)\( p T^{4} - \)\(84\!\cdots\!20\)\( p^{2} T^{5} + \)\(11\!\cdots\!40\)\( p^{3} T^{6} - \)\(84\!\cdots\!20\)\( p^{73} T^{7} + \)\(51\!\cdots\!11\)\( p^{143} T^{8} - \)\(25\!\cdots\!80\)\( p^{213} T^{9} + \)\(27\!\cdots\!90\)\( p^{284} T^{10} - \)\(20\!\cdots\!60\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 41 | \( 1 + \)\(29\!\cdots\!48\)\( T + \)\(15\!\cdots\!06\)\( T^{2} + \)\(96\!\cdots\!80\)\( p T^{3} + \)\(68\!\cdots\!95\)\( p^{2} T^{4} + \)\(32\!\cdots\!48\)\( p^{3} T^{5} + \)\(17\!\cdots\!64\)\( p^{4} T^{6} + \)\(32\!\cdots\!48\)\( p^{74} T^{7} + \)\(68\!\cdots\!95\)\( p^{144} T^{8} + \)\(96\!\cdots\!80\)\( p^{214} T^{9} + \)\(15\!\cdots\!06\)\( p^{284} T^{10} + \)\(29\!\cdots\!48\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 43 | \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(48\!\cdots\!50\)\( T^{2} + \)\(15\!\cdots\!00\)\( p T^{3} + \)\(48\!\cdots\!03\)\( p^{2} T^{4} + \)\(12\!\cdots\!00\)\( p^{3} T^{5} + \)\(29\!\cdots\!00\)\( p^{4} T^{6} + \)\(12\!\cdots\!00\)\( p^{74} T^{7} + \)\(48\!\cdots\!03\)\( p^{144} T^{8} + \)\(15\!\cdots\!00\)\( p^{214} T^{9} + \)\(48\!\cdots\!50\)\( p^{284} T^{10} + \)\(20\!\cdots\!00\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 47 | \( 1 - \)\(80\!\cdots\!80\)\( T + \)\(10\!\cdots\!10\)\( p T^{2} - \)\(87\!\cdots\!60\)\( p^{2} T^{3} + \)\(65\!\cdots\!49\)\( p^{3} T^{4} - \)\(39\!\cdots\!40\)\( p^{4} T^{5} + \)\(21\!\cdots\!80\)\( p^{5} T^{6} - \)\(39\!\cdots\!40\)\( p^{75} T^{7} + \)\(65\!\cdots\!49\)\( p^{145} T^{8} - \)\(87\!\cdots\!60\)\( p^{215} T^{9} + \)\(10\!\cdots\!10\)\( p^{285} T^{10} - \)\(80\!\cdots\!80\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 53 | \( 1 + \)\(25\!\cdots\!60\)\( T + \)\(27\!\cdots\!90\)\( p T^{2} + \)\(10\!\cdots\!80\)\( p^{2} T^{3} + \)\(61\!\cdots\!51\)\( p^{3} T^{4} + \)\(18\!\cdots\!80\)\( p^{4} T^{5} + \)\(75\!\cdots\!20\)\( p^{5} T^{6} + \)\(18\!\cdots\!80\)\( p^{75} T^{7} + \)\(61\!\cdots\!51\)\( p^{145} T^{8} + \)\(10\!\cdots\!80\)\( p^{215} T^{9} + \)\(27\!\cdots\!90\)\( p^{285} T^{10} + \)\(25\!\cdots\!60\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 59 | \( 1 - \)\(43\!\cdots\!40\)\( p T + \)\(27\!\cdots\!26\)\( p^{3} T^{2} - \)\(38\!\cdots\!00\)\( p^{3} T^{3} + \)\(79\!\cdots\!15\)\( p^{4} T^{4} - \)\(12\!\cdots\!00\)\( p^{5} T^{5} + \)\(17\!\cdots\!80\)\( p^{6} T^{6} - \)\(12\!\cdots\!00\)\( p^{76} T^{7} + \)\(79\!\cdots\!15\)\( p^{146} T^{8} - \)\(38\!\cdots\!00\)\( p^{216} T^{9} + \)\(27\!\cdots\!26\)\( p^{287} T^{10} - \)\(43\!\cdots\!40\)\( p^{356} T^{11} + p^{426} T^{12} \) | |
| 61 | \( 1 - \)\(47\!\cdots\!72\)\( T + \)\(21\!\cdots\!66\)\( p T^{2} - \)\(69\!\cdots\!20\)\( p^{2} T^{3} + \)\(47\!\cdots\!95\)\( p^{3} T^{4} - \)\(20\!\cdots\!12\)\( p^{4} T^{5} + \)\(77\!\cdots\!44\)\( p^{5} T^{6} - \)\(20\!\cdots\!12\)\( p^{75} T^{7} + \)\(47\!\cdots\!95\)\( p^{145} T^{8} - \)\(69\!\cdots\!20\)\( p^{215} T^{9} + \)\(21\!\cdots\!66\)\( p^{285} T^{10} - \)\(47\!\cdots\!72\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 67 | \( 1 + \)\(21\!\cdots\!40\)\( p T + \)\(51\!\cdots\!70\)\( p^{2} T^{2} + \)\(69\!\cdots\!80\)\( p^{3} T^{3} + \)\(93\!\cdots\!27\)\( p^{4} T^{4} + \)\(97\!\cdots\!20\)\( p^{5} T^{5} + \)\(10\!\cdots\!60\)\( p^{6} T^{6} + \)\(97\!\cdots\!20\)\( p^{76} T^{7} + \)\(93\!\cdots\!27\)\( p^{146} T^{8} + \)\(69\!\cdots\!80\)\( p^{216} T^{9} + \)\(51\!\cdots\!70\)\( p^{286} T^{10} + \)\(21\!\cdots\!40\)\( p^{356} T^{11} + p^{426} T^{12} \) | |
| 71 | \( 1 - \)\(12\!\cdots\!32\)\( T + \)\(11\!\cdots\!86\)\( T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(31\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!64\)\( T^{6} - \)\(31\!\cdots\!92\)\( p^{71} T^{7} + \)\(53\!\cdots\!95\)\( p^{142} T^{8} - \)\(85\!\cdots\!20\)\( p^{213} T^{9} + \)\(11\!\cdots\!86\)\( p^{284} T^{10} - \)\(12\!\cdots\!32\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 73 | \( 1 - \)\(23\!\cdots\!20\)\( T + \)\(10\!\cdots\!10\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!87\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!80\)\( T^{6} - \)\(40\!\cdots\!60\)\( p^{71} T^{7} + \)\(39\!\cdots\!87\)\( p^{142} T^{8} - \)\(14\!\cdots\!40\)\( p^{213} T^{9} + \)\(10\!\cdots\!10\)\( p^{284} T^{10} - \)\(23\!\cdots\!20\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 79 | \( 1 + \)\(60\!\cdots\!20\)\( T + \)\(30\!\cdots\!74\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!15\)\( T^{4} + \)\(35\!\cdots\!00\)\( T^{5} + \)\(91\!\cdots\!80\)\( T^{6} + \)\(35\!\cdots\!00\)\( p^{71} T^{7} + \)\(22\!\cdots\!15\)\( p^{142} T^{8} + \)\(83\!\cdots\!00\)\( p^{213} T^{9} + \)\(30\!\cdots\!74\)\( p^{284} T^{10} + \)\(60\!\cdots\!20\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 83 | \( 1 - \)\(47\!\cdots\!60\)\( T + \)\(18\!\cdots\!30\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!67\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!40\)\( T^{6} - \)\(17\!\cdots\!80\)\( p^{71} T^{7} + \)\(10\!\cdots\!67\)\( p^{142} T^{8} - \)\(46\!\cdots\!20\)\( p^{213} T^{9} + \)\(18\!\cdots\!30\)\( p^{284} T^{10} - \)\(47\!\cdots\!60\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 89 | \( 1 + \)\(47\!\cdots\!60\)\( T + \)\(20\!\cdots\!34\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(40\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!00\)\( p^{71} T^{7} + \)\(12\!\cdots\!15\)\( p^{142} T^{8} + \)\(52\!\cdots\!00\)\( p^{213} T^{9} + \)\(20\!\cdots\!34\)\( p^{284} T^{10} + \)\(47\!\cdots\!60\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 97 | \( 1 - \)\(90\!\cdots\!80\)\( T + \)\(81\!\cdots\!70\)\( T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!27\)\( T^{4} - \)\(85\!\cdots\!40\)\( T^{5} + \)\(32\!\cdots\!60\)\( T^{6} - \)\(85\!\cdots\!40\)\( p^{71} T^{7} + \)\(22\!\cdots\!27\)\( p^{142} T^{8} - \)\(43\!\cdots\!40\)\( p^{213} T^{9} + \)\(81\!\cdots\!70\)\( p^{284} T^{10} - \)\(90\!\cdots\!80\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.39451565079538353000149096907, −7.32383613248478915281172666756, −6.54386990840114140755872850028, −6.39152354885930146537147423028, −5.99354682246809962928840741640, −5.75502253196001092456127417807, −5.66795524891022245294555057796, −4.96037145183897123927359259079, −4.82425745080437901523267684343, −4.74886003849360771254735722472, −4.19594247914254030341103452850, −4.11342627530132691716016627456, −3.97489418947681100031575860786, −3.48746634772373262871765394213, −3.33263817210924801839587871020, −2.89963734039042422857166593637, −2.78103252250850026748902801876, −2.18585290000314499876701611719, −2.17928549147365687300460926319, −2.04895236697919141331455380395, −1.30398757314287893290970702072, −1.06667589081620936244469705442, −0.65951141890227211426020692908, −0.58152237278868291228749926133, −0.00194874992571728991050332610, 0.00194874992571728991050332610, 0.58152237278868291228749926133, 0.65951141890227211426020692908, 1.06667589081620936244469705442, 1.30398757314287893290970702072, 2.04895236697919141331455380395, 2.17928549147365687300460926319, 2.18585290000314499876701611719, 2.78103252250850026748902801876, 2.89963734039042422857166593637, 3.33263817210924801839587871020, 3.48746634772373262871765394213, 3.97489418947681100031575860786, 4.11342627530132691716016627456, 4.19594247914254030341103452850, 4.74886003849360771254735722472, 4.82425745080437901523267684343, 4.96037145183897123927359259079, 5.66795524891022245294555057796, 5.75502253196001092456127417807, 5.99354682246809962928840741640, 6.39152354885930146537147423028, 6.54386990840114140755872850028, 7.32383613248478915281172666756, 7.39451565079538353000149096907
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.