Dirichlet series
| L(s) = 1 | + 6.21e9·2-s + 1.11e16·3-s − 4.57e19·4-s + 3.51e22·5-s + 6.90e25·6-s + 1.31e27·7-s − 2.02e29·8-s + 7.21e31·9-s + 2.18e32·10-s − 2.32e33·11-s − 5.08e35·12-s + 9.61e35·13-s + 8.18e36·14-s + 3.90e38·15-s + 2.03e39·16-s − 1.32e40·17-s + 4.47e41·18-s + 5.60e41·19-s − 1.60e42·20-s + 1.46e43·21-s − 1.44e43·22-s + 4.78e43·23-s − 2.25e45·24-s − 4.56e45·25-s + 5.96e45·26-s + 3.56e47·27-s − 6.02e46·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 3.46·3-s − 1.23·4-s + 0.675·5-s + 3.54·6-s + 0.450·7-s − 0.904·8-s + 7·9-s + 0.690·10-s − 0.331·11-s − 4.29·12-s + 0.602·13-s + 0.461·14-s + 2.33·15-s + 1.49·16-s − 1.36·17-s + 7.15·18-s + 1.54·19-s − 0.836·20-s + 1.56·21-s − 0.338·22-s + 0.265·23-s − 3.13·24-s − 1.68·25-s + 0.615·26-s + 10.7·27-s − 0.558·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+65/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(729\) = \(3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.67531\times 10^{11}\) |
| Root analytic conductor: | \(8.95944\) |
| Motivic weight: | \(65\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 729,\ (\ :[65/2]^{6}),\ 1)\) |
Particular Values
| \(L(33)\) | \(\approx\) | \(0.8768175360\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8768175360\) |
| \(L(\frac{67}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p^{32} T )^{6} \) |
| good | 2 | \( 1 - 3105491481 p T + 2634317705913816507 p^{5} T^{2} - \)\(73\!\cdots\!93\)\( p^{13} T^{3} + \)\(32\!\cdots\!09\)\( p^{27} T^{4} - \)\(39\!\cdots\!43\)\( p^{43} T^{5} + \)\(78\!\cdots\!07\)\( p^{61} T^{6} - \)\(39\!\cdots\!43\)\( p^{108} T^{7} + \)\(32\!\cdots\!09\)\( p^{157} T^{8} - \)\(73\!\cdots\!93\)\( p^{208} T^{9} + 2634317705913816507 p^{265} T^{10} - 3105491481 p^{326} T^{11} + p^{390} T^{12} \) |
| 5 | \( 1 - \)\(14\!\cdots\!68\)\( p^{2} T + \)\(92\!\cdots\!06\)\( p^{4} T^{2} - \)\(17\!\cdots\!84\)\( p^{6} T^{3} + \)\(11\!\cdots\!19\)\( p^{12} T^{4} - \)\(24\!\cdots\!44\)\( p^{21} T^{5} + \)\(17\!\cdots\!08\)\( p^{31} T^{6} - \)\(24\!\cdots\!44\)\( p^{86} T^{7} + \)\(11\!\cdots\!19\)\( p^{142} T^{8} - \)\(17\!\cdots\!84\)\( p^{201} T^{9} + \)\(92\!\cdots\!06\)\( p^{264} T^{10} - \)\(14\!\cdots\!68\)\( p^{327} T^{11} + p^{390} T^{12} \) | |
| 7 | \( 1 - \)\(13\!\cdots\!04\)\( T + \)\(67\!\cdots\!06\)\( p^{2} T^{2} - \)\(25\!\cdots\!36\)\( p^{5} T^{3} + \)\(28\!\cdots\!77\)\( p^{11} T^{4} - \)\(25\!\cdots\!16\)\( p^{17} T^{5} + \)\(43\!\cdots\!04\)\( p^{25} T^{6} - \)\(25\!\cdots\!16\)\( p^{82} T^{7} + \)\(28\!\cdots\!77\)\( p^{141} T^{8} - \)\(25\!\cdots\!36\)\( p^{200} T^{9} + \)\(67\!\cdots\!06\)\( p^{262} T^{10} - \)\(13\!\cdots\!04\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 11 | \( 1 + \)\(23\!\cdots\!76\)\( T + \)\(30\!\cdots\!74\)\( p^{3} T^{2} + \)\(21\!\cdots\!52\)\( p^{4} T^{3} + \)\(88\!\cdots\!03\)\( p^{6} T^{4} - \)\(14\!\cdots\!60\)\( p^{11} T^{5} + \)\(24\!\cdots\!68\)\( p^{16} T^{6} - \)\(14\!\cdots\!60\)\( p^{76} T^{7} + \)\(88\!\cdots\!03\)\( p^{136} T^{8} + \)\(21\!\cdots\!52\)\( p^{199} T^{9} + \)\(30\!\cdots\!74\)\( p^{263} T^{10} + \)\(23\!\cdots\!76\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 13 | \( 1 - \)\(96\!\cdots\!48\)\( T + \)\(69\!\cdots\!90\)\( p T^{2} - \)\(40\!\cdots\!92\)\( p^{3} T^{3} + \)\(12\!\cdots\!75\)\( p^{5} T^{4} - \)\(63\!\cdots\!84\)\( p^{7} T^{5} + \)\(10\!\cdots\!04\)\( p^{10} T^{6} - \)\(63\!\cdots\!84\)\( p^{72} T^{7} + \)\(12\!\cdots\!75\)\( p^{135} T^{8} - \)\(40\!\cdots\!92\)\( p^{198} T^{9} + \)\(69\!\cdots\!90\)\( p^{261} T^{10} - \)\(96\!\cdots\!48\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 17 | \( 1 + \)\(13\!\cdots\!32\)\( T + \)\(61\!\cdots\!06\)\( p T^{2} + \)\(44\!\cdots\!20\)\( p^{2} T^{3} + \)\(18\!\cdots\!55\)\( p^{4} T^{4} + \)\(42\!\cdots\!84\)\( p^{7} T^{5} + \)\(10\!\cdots\!36\)\( p^{10} T^{6} + \)\(42\!\cdots\!84\)\( p^{72} T^{7} + \)\(18\!\cdots\!55\)\( p^{134} T^{8} + \)\(44\!\cdots\!20\)\( p^{197} T^{9} + \)\(61\!\cdots\!06\)\( p^{261} T^{10} + \)\(13\!\cdots\!32\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 19 | \( 1 - \)\(29\!\cdots\!68\)\( p T + \)\(15\!\cdots\!38\)\( p T^{2} - \)\(12\!\cdots\!64\)\( p^{2} T^{3} + \)\(98\!\cdots\!83\)\( p^{4} T^{4} + \)\(31\!\cdots\!48\)\( p^{7} T^{5} + \)\(15\!\cdots\!68\)\( p^{10} T^{6} + \)\(31\!\cdots\!48\)\( p^{72} T^{7} + \)\(98\!\cdots\!83\)\( p^{134} T^{8} - \)\(12\!\cdots\!64\)\( p^{197} T^{9} + \)\(15\!\cdots\!38\)\( p^{261} T^{10} - \)\(29\!\cdots\!68\)\( p^{326} T^{11} + p^{390} T^{12} \) | |
| 23 | \( 1 - \)\(47\!\cdots\!92\)\( T + \)\(10\!\cdots\!86\)\( T^{2} + \)\(93\!\cdots\!72\)\( p T^{3} + \)\(44\!\cdots\!73\)\( p^{3} T^{4} + \)\(30\!\cdots\!48\)\( p^{5} T^{5} + \)\(61\!\cdots\!96\)\( p^{7} T^{6} + \)\(30\!\cdots\!48\)\( p^{70} T^{7} + \)\(44\!\cdots\!73\)\( p^{133} T^{8} + \)\(93\!\cdots\!72\)\( p^{196} T^{9} + \)\(10\!\cdots\!86\)\( p^{260} T^{10} - \)\(47\!\cdots\!92\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 29 | \( 1 - \)\(18\!\cdots\!04\)\( T + \)\(15\!\cdots\!94\)\( p T^{2} - \)\(84\!\cdots\!92\)\( p^{2} T^{3} + \)\(13\!\cdots\!67\)\( p^{4} T^{4} - \)\(20\!\cdots\!84\)\( p^{6} T^{5} + \)\(26\!\cdots\!72\)\( p^{8} T^{6} - \)\(20\!\cdots\!84\)\( p^{71} T^{7} + \)\(13\!\cdots\!67\)\( p^{134} T^{8} - \)\(84\!\cdots\!92\)\( p^{197} T^{9} + \)\(15\!\cdots\!94\)\( p^{261} T^{10} - \)\(18\!\cdots\!04\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 31 | \( 1 - \)\(26\!\cdots\!96\)\( T + \)\(10\!\cdots\!38\)\( p T^{2} - \)\(10\!\cdots\!68\)\( p^{2} T^{3} + \)\(61\!\cdots\!43\)\( p^{4} T^{4} - \)\(18\!\cdots\!44\)\( p^{6} T^{5} + \)\(71\!\cdots\!72\)\( p^{8} T^{6} - \)\(18\!\cdots\!44\)\( p^{71} T^{7} + \)\(61\!\cdots\!43\)\( p^{134} T^{8} - \)\(10\!\cdots\!68\)\( p^{197} T^{9} + \)\(10\!\cdots\!38\)\( p^{261} T^{10} - \)\(26\!\cdots\!96\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 37 | \( 1 + \)\(19\!\cdots\!76\)\( T + \)\(53\!\cdots\!74\)\( T^{2} + \)\(68\!\cdots\!08\)\( T^{3} + \)\(29\!\cdots\!43\)\( p T^{4} + \)\(77\!\cdots\!92\)\( p^{2} T^{5} + \)\(24\!\cdots\!36\)\( p^{3} T^{6} + \)\(77\!\cdots\!92\)\( p^{67} T^{7} + \)\(29\!\cdots\!43\)\( p^{131} T^{8} + \)\(68\!\cdots\!08\)\( p^{195} T^{9} + \)\(53\!\cdots\!74\)\( p^{260} T^{10} + \)\(19\!\cdots\!76\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 41 | \( 1 - \)\(44\!\cdots\!16\)\( T + \)\(38\!\cdots\!78\)\( T^{2} - \)\(29\!\cdots\!88\)\( p T^{3} + \)\(34\!\cdots\!43\)\( p^{2} T^{4} - \)\(19\!\cdots\!64\)\( p^{3} T^{5} + \)\(17\!\cdots\!92\)\( p^{4} T^{6} - \)\(19\!\cdots\!64\)\( p^{68} T^{7} + \)\(34\!\cdots\!43\)\( p^{132} T^{8} - \)\(29\!\cdots\!88\)\( p^{196} T^{9} + \)\(38\!\cdots\!78\)\( p^{260} T^{10} - \)\(44\!\cdots\!16\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 43 | \( 1 - \)\(15\!\cdots\!40\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(14\!\cdots\!40\)\( p T^{3} + \)\(87\!\cdots\!03\)\( p^{2} T^{4} - \)\(19\!\cdots\!60\)\( p^{3} T^{5} + \)\(89\!\cdots\!60\)\( p^{4} T^{6} - \)\(19\!\cdots\!60\)\( p^{68} T^{7} + \)\(87\!\cdots\!03\)\( p^{132} T^{8} - \)\(14\!\cdots\!40\)\( p^{196} T^{9} + \)\(57\!\cdots\!70\)\( p^{260} T^{10} - \)\(15\!\cdots\!40\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 47 | \( 1 + \)\(23\!\cdots\!80\)\( T + \)\(24\!\cdots\!90\)\( p T^{2} + \)\(40\!\cdots\!40\)\( p^{2} T^{3} + \)\(21\!\cdots\!89\)\( p^{3} T^{4} - \)\(15\!\cdots\!60\)\( p^{4} T^{5} - \)\(35\!\cdots\!80\)\( p^{5} T^{6} - \)\(15\!\cdots\!60\)\( p^{69} T^{7} + \)\(21\!\cdots\!89\)\( p^{133} T^{8} + \)\(40\!\cdots\!40\)\( p^{197} T^{9} + \)\(24\!\cdots\!90\)\( p^{261} T^{10} + \)\(23\!\cdots\!80\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 53 | \( 1 - \)\(96\!\cdots\!32\)\( T + \)\(97\!\cdots\!42\)\( p T^{2} - \)\(11\!\cdots\!16\)\( p^{2} T^{3} + \)\(82\!\cdots\!83\)\( p^{3} T^{4} - \)\(79\!\cdots\!56\)\( p^{4} T^{5} + \)\(43\!\cdots\!44\)\( p^{5} T^{6} - \)\(79\!\cdots\!56\)\( p^{69} T^{7} + \)\(82\!\cdots\!83\)\( p^{133} T^{8} - \)\(11\!\cdots\!16\)\( p^{197} T^{9} + \)\(97\!\cdots\!42\)\( p^{261} T^{10} - \)\(96\!\cdots\!32\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 59 | \( 1 + \)\(12\!\cdots\!28\)\( p T + \)\(19\!\cdots\!62\)\( p^{2} T^{2} + \)\(18\!\cdots\!56\)\( p^{3} T^{3} + \)\(17\!\cdots\!83\)\( p^{4} T^{4} + \)\(12\!\cdots\!32\)\( p^{5} T^{5} + \)\(83\!\cdots\!08\)\( p^{6} T^{6} + \)\(12\!\cdots\!32\)\( p^{70} T^{7} + \)\(17\!\cdots\!83\)\( p^{134} T^{8} + \)\(18\!\cdots\!56\)\( p^{198} T^{9} + \)\(19\!\cdots\!62\)\( p^{262} T^{10} + \)\(12\!\cdots\!28\)\( p^{326} T^{11} + p^{390} T^{12} \) | |
| 61 | \( 1 - \)\(40\!\cdots\!00\)\( T + \)\(97\!\cdots\!74\)\( p T^{2} - \)\(47\!\cdots\!00\)\( p^{2} T^{3} + \)\(67\!\cdots\!35\)\( p^{3} T^{4} - \)\(24\!\cdots\!00\)\( p^{4} T^{5} + \)\(25\!\cdots\!00\)\( p^{5} T^{6} - \)\(24\!\cdots\!00\)\( p^{69} T^{7} + \)\(67\!\cdots\!35\)\( p^{133} T^{8} - \)\(47\!\cdots\!00\)\( p^{197} T^{9} + \)\(97\!\cdots\!74\)\( p^{261} T^{10} - \)\(40\!\cdots\!00\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 67 | \( 1 - \)\(14\!\cdots\!08\)\( T + \)\(22\!\cdots\!02\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!55\)\( T^{4} - \)\(24\!\cdots\!68\)\( T^{5} + \)\(14\!\cdots\!84\)\( T^{6} - \)\(24\!\cdots\!68\)\( p^{65} T^{7} + \)\(23\!\cdots\!55\)\( p^{130} T^{8} - \)\(28\!\cdots\!20\)\( p^{195} T^{9} + \)\(22\!\cdots\!02\)\( p^{260} T^{10} - \)\(14\!\cdots\!08\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 71 | \( 1 - \)\(64\!\cdots\!72\)\( p T + \)\(14\!\cdots\!66\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(75\!\cdots\!92\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(75\!\cdots\!92\)\( p^{65} T^{7} + \)\(53\!\cdots\!95\)\( p^{130} T^{8} - \)\(30\!\cdots\!20\)\( p^{195} T^{9} + \)\(14\!\cdots\!66\)\( p^{260} T^{10} - \)\(64\!\cdots\!72\)\( p^{326} T^{11} + p^{390} T^{12} \) | |
| 73 | \( 1 - \)\(52\!\cdots\!52\)\( T + \)\(75\!\cdots\!26\)\( T^{2} - \)\(31\!\cdots\!84\)\( T^{3} + \)\(24\!\cdots\!31\)\( T^{4} - \)\(79\!\cdots\!96\)\( T^{5} + \)\(41\!\cdots\!32\)\( T^{6} - \)\(79\!\cdots\!96\)\( p^{65} T^{7} + \)\(24\!\cdots\!31\)\( p^{130} T^{8} - \)\(31\!\cdots\!84\)\( p^{195} T^{9} + \)\(75\!\cdots\!26\)\( p^{260} T^{10} - \)\(52\!\cdots\!52\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 79 | \( 1 - \)\(54\!\cdots\!40\)\( T + \)\(10\!\cdots\!94\)\( T^{2} - \)\(51\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} - \)\(20\!\cdots\!00\)\( p^{65} T^{7} + \)\(55\!\cdots\!15\)\( p^{130} T^{8} - \)\(51\!\cdots\!00\)\( p^{195} T^{9} + \)\(10\!\cdots\!94\)\( p^{260} T^{10} - \)\(54\!\cdots\!40\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 83 | \( 1 - \)\(12\!\cdots\!56\)\( T + \)\(88\!\cdots\!26\)\( T^{2} - \)\(43\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!67\)\( T^{4} - \)\(50\!\cdots\!92\)\( T^{5} + \)\(13\!\cdots\!36\)\( T^{6} - \)\(50\!\cdots\!92\)\( p^{65} T^{7} + \)\(16\!\cdots\!67\)\( p^{130} T^{8} - \)\(43\!\cdots\!72\)\( p^{195} T^{9} + \)\(88\!\cdots\!26\)\( p^{260} T^{10} - \)\(12\!\cdots\!56\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 89 | \( 1 - \)\(64\!\cdots\!12\)\( T + \)\(35\!\cdots\!62\)\( T^{2} - \)\(12\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!63\)\( T^{4} - \)\(96\!\cdots\!48\)\( T^{5} + \)\(24\!\cdots\!48\)\( T^{6} - \)\(96\!\cdots\!48\)\( p^{65} T^{7} + \)\(39\!\cdots\!63\)\( p^{130} T^{8} - \)\(12\!\cdots\!64\)\( p^{195} T^{9} + \)\(35\!\cdots\!62\)\( p^{260} T^{10} - \)\(64\!\cdots\!12\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
| 97 | \( 1 - \)\(12\!\cdots\!28\)\( T + \)\(12\!\cdots\!02\)\( T^{2} - \)\(86\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!55\)\( T^{4} - \)\(23\!\cdots\!68\)\( T^{5} + \)\(95\!\cdots\!44\)\( T^{6} - \)\(23\!\cdots\!68\)\( p^{65} T^{7} + \)\(50\!\cdots\!55\)\( p^{130} T^{8} - \)\(86\!\cdots\!20\)\( p^{195} T^{9} + \)\(12\!\cdots\!02\)\( p^{260} T^{10} - \)\(12\!\cdots\!28\)\( p^{325} T^{11} + p^{390} T^{12} \) | |
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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
−5.92979394939576211699525347524, −5.25129037306274752168150383051, −4.96209925583452235025950003711, −4.90969088114051496638591555968, −4.87336752232567447310001262839, −4.58245597595974450965292405407, −4.58095151476954735032538256490, −3.92989389927962787383664339207, −3.74950179448507353652194306493, −3.67133610237072394391984520135, −3.43890774386783786363461435079, −3.41945440576368461067653105517, −3.32062737752976500789271382664, −2.52738967925805034837650733717, −2.45458821858320283230936286468, −2.39107167226987698435720894102, −2.33340319561559886548535176232, −1.75778697562675250265946578206, −1.69367592205243487771671221798, −1.68538023806704416717028753026, −0.994545671737636078046059527338, −0.947411672870239258578513994100, −0.917066031967248182833924348796, −0.56352291936085473274108862816, −0.02160325245556711689733982665, 0.02160325245556711689733982665, 0.56352291936085473274108862816, 0.917066031967248182833924348796, 0.947411672870239258578513994100, 0.994545671737636078046059527338, 1.68538023806704416717028753026, 1.69367592205243487771671221798, 1.75778697562675250265946578206, 2.33340319561559886548535176232, 2.39107167226987698435720894102, 2.45458821858320283230936286468, 2.52738967925805034837650733717, 3.32062737752976500789271382664, 3.41945440576368461067653105517, 3.43890774386783786363461435079, 3.67133610237072394391984520135, 3.74950179448507353652194306493, 3.92989389927962787383664339207, 4.58095151476954735032538256490, 4.58245597595974450965292405407, 4.87336752232567447310001262839, 4.90969088114051496638591555968, 4.96209925583452235025950003711, 5.25129037306274752168150383051, 5.92979394939576211699525347524
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.