Dirichlet series
| L(s) = 1 | − 8.69e8·2-s − 1.00e17·3-s − 1.08e21·4-s + 5.36e23·5-s + 8.69e25·6-s − 1.65e29·7-s + 1.07e31·8-s + 5.84e33·9-s − 4.66e32·10-s − 1.92e36·11-s + 1.08e38·12-s − 4.72e37·13-s + 1.43e38·14-s − 5.37e40·15-s + 4.71e41·16-s − 5.05e42·17-s − 5.07e42·18-s − 6.58e43·19-s − 5.82e44·20-s + 1.65e46·21-s + 1.66e45·22-s − 5.13e46·23-s − 1.07e48·24-s − 4.26e48·25-s + 4.10e46·26-s − 2.59e50·27-s + 1.79e50·28-s + ⋯ |
| L(s) = 1 | − 0.0357·2-s − 3.46·3-s − 1.83·4-s + 0.412·5-s + 0.123·6-s − 1.15·7-s + 0.752·8-s + 7·9-s − 0.0147·10-s − 2.26·11-s + 6.36·12-s − 0.175·13-s + 0.0413·14-s − 1.42·15-s + 1.35·16-s − 1.79·17-s − 0.250·18-s − 0.502·19-s − 0.757·20-s + 4.00·21-s + 0.0811·22-s − 0.537·23-s − 2.60·24-s − 2.51·25-s + 0.00626·26-s − 10.7·27-s + 2.12·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(70-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+69/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(729\) = \(3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.47747\times 10^{11}\) |
| Root analytic conductor: | \(9.51075\) |
| Motivic weight: | \(69\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 729,\ (\ :[69/2]^{6}),\ 1)\) |
Particular Values
| \(L(35)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{71}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{34} T )^{6} \) |
| good | 2 | \( 1 + 217340847 p^{2} T + 4240847682625263159 p^{8} T^{2} - \)\(33\!\cdots\!49\)\( p^{18} T^{3} + \)\(40\!\cdots\!13\)\( p^{34} T^{4} - \)\(31\!\cdots\!79\)\( p^{52} T^{5} + \)\(92\!\cdots\!59\)\( p^{72} T^{6} - \)\(31\!\cdots\!79\)\( p^{121} T^{7} + \)\(40\!\cdots\!13\)\( p^{172} T^{8} - \)\(33\!\cdots\!49\)\( p^{225} T^{9} + 4240847682625263159 p^{284} T^{10} + 217340847 p^{347} T^{11} + p^{414} T^{12} \) |
| 5 | \( 1 - \)\(10\!\cdots\!04\)\( p T + \)\(72\!\cdots\!14\)\( p^{4} T^{2} - \)\(97\!\cdots\!48\)\( p^{8} T^{3} + \)\(67\!\cdots\!51\)\( p^{13} T^{4} - \)\(22\!\cdots\!12\)\( p^{21} T^{5} + \)\(24\!\cdots\!72\)\( p^{31} T^{6} - \)\(22\!\cdots\!12\)\( p^{90} T^{7} + \)\(67\!\cdots\!51\)\( p^{151} T^{8} - \)\(97\!\cdots\!48\)\( p^{215} T^{9} + \)\(72\!\cdots\!14\)\( p^{280} T^{10} - \)\(10\!\cdots\!04\)\( p^{346} T^{11} + p^{414} T^{12} \) | |
| 7 | \( 1 + \)\(23\!\cdots\!88\)\( p T + \)\(11\!\cdots\!46\)\( p^{2} T^{2} + \)\(13\!\cdots\!36\)\( p^{7} T^{3} + \)\(14\!\cdots\!11\)\( p^{12} T^{4} + \)\(23\!\cdots\!16\)\( p^{19} T^{5} + \)\(76\!\cdots\!36\)\( p^{27} T^{6} + \)\(23\!\cdots\!16\)\( p^{88} T^{7} + \)\(14\!\cdots\!11\)\( p^{150} T^{8} + \)\(13\!\cdots\!36\)\( p^{214} T^{9} + \)\(11\!\cdots\!46\)\( p^{278} T^{10} + \)\(23\!\cdots\!88\)\( p^{346} T^{11} + p^{414} T^{12} \) | |
| 11 | \( 1 + \)\(19\!\cdots\!76\)\( T + \)\(21\!\cdots\!34\)\( p T^{2} + \)\(68\!\cdots\!12\)\( p^{4} T^{3} + \)\(40\!\cdots\!03\)\( p^{8} T^{4} - \)\(21\!\cdots\!20\)\( p^{12} T^{5} - \)\(84\!\cdots\!72\)\( p^{16} T^{6} - \)\(21\!\cdots\!20\)\( p^{81} T^{7} + \)\(40\!\cdots\!03\)\( p^{146} T^{8} + \)\(68\!\cdots\!12\)\( p^{211} T^{9} + \)\(21\!\cdots\!34\)\( p^{277} T^{10} + \)\(19\!\cdots\!76\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 13 | \( 1 + \)\(47\!\cdots\!52\)\( T + \)\(21\!\cdots\!70\)\( p T^{2} + \)\(16\!\cdots\!48\)\( p^{3} T^{3} + \)\(74\!\cdots\!95\)\( p^{6} T^{4} + \)\(47\!\cdots\!28\)\( p^{10} T^{5} + \)\(75\!\cdots\!44\)\( p^{14} T^{6} + \)\(47\!\cdots\!28\)\( p^{79} T^{7} + \)\(74\!\cdots\!95\)\( p^{144} T^{8} + \)\(16\!\cdots\!48\)\( p^{210} T^{9} + \)\(21\!\cdots\!70\)\( p^{277} T^{10} + \)\(47\!\cdots\!52\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 17 | \( 1 + \)\(29\!\cdots\!36\)\( p T + \)\(88\!\cdots\!78\)\( p^{2} T^{2} + \)\(47\!\cdots\!80\)\( p^{4} T^{3} + \)\(55\!\cdots\!95\)\( p^{6} T^{4} + \)\(75\!\cdots\!52\)\( p^{8} T^{5} + \)\(47\!\cdots\!76\)\( p^{10} T^{6} + \)\(75\!\cdots\!52\)\( p^{77} T^{7} + \)\(55\!\cdots\!95\)\( p^{144} T^{8} + \)\(47\!\cdots\!80\)\( p^{211} T^{9} + \)\(88\!\cdots\!78\)\( p^{278} T^{10} + \)\(29\!\cdots\!36\)\( p^{346} T^{11} + p^{414} T^{12} \) | |
| 19 | \( 1 + \)\(65\!\cdots\!88\)\( T + \)\(28\!\cdots\!98\)\( p T^{2} + \)\(27\!\cdots\!16\)\( p^{2} T^{3} + \)\(94\!\cdots\!03\)\( p^{4} T^{4} - \)\(37\!\cdots\!52\)\( p^{7} T^{5} + \)\(17\!\cdots\!92\)\( p^{11} T^{6} - \)\(37\!\cdots\!52\)\( p^{76} T^{7} + \)\(94\!\cdots\!03\)\( p^{142} T^{8} + \)\(27\!\cdots\!16\)\( p^{209} T^{9} + \)\(28\!\cdots\!98\)\( p^{277} T^{10} + \)\(65\!\cdots\!88\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 23 | \( 1 + \)\(22\!\cdots\!56\)\( p T + \)\(57\!\cdots\!14\)\( p^{2} T^{2} + \)\(89\!\cdots\!48\)\( p^{3} T^{3} + \)\(16\!\cdots\!31\)\( p^{4} T^{4} + \)\(23\!\cdots\!48\)\( p^{5} T^{5} + \)\(33\!\cdots\!08\)\( p^{6} T^{6} + \)\(23\!\cdots\!48\)\( p^{74} T^{7} + \)\(16\!\cdots\!31\)\( p^{142} T^{8} + \)\(89\!\cdots\!48\)\( p^{210} T^{9} + \)\(57\!\cdots\!14\)\( p^{278} T^{10} + \)\(22\!\cdots\!56\)\( p^{346} T^{11} + p^{414} T^{12} \) | |
| 29 | \( 1 - \)\(78\!\cdots\!24\)\( T + \)\(18\!\cdots\!14\)\( p T^{2} - \)\(30\!\cdots\!12\)\( p^{2} T^{3} + \)\(46\!\cdots\!63\)\( p^{3} T^{4} - \)\(18\!\cdots\!76\)\( p^{5} T^{5} + \)\(70\!\cdots\!28\)\( p^{7} T^{6} - \)\(18\!\cdots\!76\)\( p^{74} T^{7} + \)\(46\!\cdots\!63\)\( p^{141} T^{8} - \)\(30\!\cdots\!12\)\( p^{209} T^{9} + \)\(18\!\cdots\!14\)\( p^{277} T^{10} - \)\(78\!\cdots\!24\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 31 | \( 1 - \)\(69\!\cdots\!96\)\( T + \)\(19\!\cdots\!58\)\( p T^{2} - \)\(26\!\cdots\!68\)\( p^{2} T^{3} + \)\(13\!\cdots\!03\)\( p^{4} T^{4} - \)\(44\!\cdots\!24\)\( p^{6} T^{5} + \)\(15\!\cdots\!52\)\( p^{8} T^{6} - \)\(44\!\cdots\!24\)\( p^{75} T^{7} + \)\(13\!\cdots\!03\)\( p^{142} T^{8} - \)\(26\!\cdots\!68\)\( p^{209} T^{9} + \)\(19\!\cdots\!58\)\( p^{277} T^{10} - \)\(69\!\cdots\!96\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 37 | \( 1 + \)\(75\!\cdots\!96\)\( T + \)\(54\!\cdots\!74\)\( T^{2} + \)\(33\!\cdots\!88\)\( T^{3} + \)\(43\!\cdots\!43\)\( p T^{4} + \)\(64\!\cdots\!32\)\( p^{2} T^{5} + \)\(63\!\cdots\!96\)\( p^{3} T^{6} + \)\(64\!\cdots\!32\)\( p^{71} T^{7} + \)\(43\!\cdots\!43\)\( p^{139} T^{8} + \)\(33\!\cdots\!88\)\( p^{207} T^{9} + \)\(54\!\cdots\!74\)\( p^{276} T^{10} + \)\(75\!\cdots\!96\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 41 | \( 1 + \)\(25\!\cdots\!24\)\( T + \)\(72\!\cdots\!38\)\( T^{2} + \)\(17\!\cdots\!12\)\( T^{3} + \)\(67\!\cdots\!83\)\( p T^{4} + \)\(34\!\cdots\!96\)\( p^{2} T^{5} + \)\(94\!\cdots\!92\)\( p^{3} T^{6} + \)\(34\!\cdots\!96\)\( p^{71} T^{7} + \)\(67\!\cdots\!83\)\( p^{139} T^{8} + \)\(17\!\cdots\!12\)\( p^{207} T^{9} + \)\(72\!\cdots\!38\)\( p^{276} T^{10} + \)\(25\!\cdots\!24\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 43 | \( 1 + \)\(67\!\cdots\!40\)\( T + \)\(42\!\cdots\!10\)\( T^{2} + \)\(40\!\cdots\!40\)\( p T^{3} + \)\(33\!\cdots\!03\)\( p^{2} T^{4} + \)\(22\!\cdots\!60\)\( p^{3} T^{5} + \)\(12\!\cdots\!80\)\( p^{4} T^{6} + \)\(22\!\cdots\!60\)\( p^{72} T^{7} + \)\(33\!\cdots\!03\)\( p^{140} T^{8} + \)\(40\!\cdots\!40\)\( p^{208} T^{9} + \)\(42\!\cdots\!10\)\( p^{276} T^{10} + \)\(67\!\cdots\!40\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 47 | \( 1 - \)\(75\!\cdots\!20\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(16\!\cdots\!20\)\( p T^{3} + \)\(30\!\cdots\!63\)\( p^{2} T^{4} - \)\(32\!\cdots\!20\)\( p^{3} T^{5} + \)\(42\!\cdots\!40\)\( p^{4} T^{6} - \)\(32\!\cdots\!20\)\( p^{72} T^{7} + \)\(30\!\cdots\!63\)\( p^{140} T^{8} - \)\(16\!\cdots\!20\)\( p^{208} T^{9} + \)\(12\!\cdots\!30\)\( p^{276} T^{10} - \)\(75\!\cdots\!20\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 53 | \( 1 + \)\(44\!\cdots\!28\)\( T + \)\(48\!\cdots\!26\)\( T^{2} + \)\(28\!\cdots\!32\)\( p T^{3} + \)\(34\!\cdots\!59\)\( p^{2} T^{4} + \)\(16\!\cdots\!72\)\( p^{3} T^{5} + \)\(14\!\cdots\!32\)\( p^{4} T^{6} + \)\(16\!\cdots\!72\)\( p^{72} T^{7} + \)\(34\!\cdots\!59\)\( p^{140} T^{8} + \)\(28\!\cdots\!32\)\( p^{208} T^{9} + \)\(48\!\cdots\!26\)\( p^{276} T^{10} + \)\(44\!\cdots\!28\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 59 | \( 1 - \)\(97\!\cdots\!28\)\( T + \)\(39\!\cdots\!78\)\( p T^{2} - \)\(16\!\cdots\!36\)\( p^{2} T^{3} + \)\(12\!\cdots\!97\)\( p^{3} T^{4} - \)\(41\!\cdots\!12\)\( p^{4} T^{5} + \)\(12\!\cdots\!32\)\( p^{5} T^{6} - \)\(41\!\cdots\!12\)\( p^{73} T^{7} + \)\(12\!\cdots\!97\)\( p^{141} T^{8} - \)\(16\!\cdots\!36\)\( p^{209} T^{9} + \)\(39\!\cdots\!78\)\( p^{277} T^{10} - \)\(97\!\cdots\!28\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 61 | \( 1 + \)\(59\!\cdots\!80\)\( T + \)\(80\!\cdots\!94\)\( p T^{2} + \)\(70\!\cdots\!60\)\( p^{2} T^{3} + \)\(57\!\cdots\!95\)\( p^{3} T^{4} + \)\(44\!\cdots\!60\)\( p^{4} T^{5} + \)\(26\!\cdots\!60\)\( p^{5} T^{6} + \)\(44\!\cdots\!60\)\( p^{73} T^{7} + \)\(57\!\cdots\!95\)\( p^{141} T^{8} + \)\(70\!\cdots\!60\)\( p^{209} T^{9} + \)\(80\!\cdots\!94\)\( p^{277} T^{10} + \)\(59\!\cdots\!80\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 67 | \( 1 - \)\(19\!\cdots\!44\)\( p T + \)\(11\!\cdots\!78\)\( p^{2} T^{2} - \)\(16\!\cdots\!40\)\( p^{3} T^{3} + \)\(57\!\cdots\!55\)\( p^{4} T^{4} - \)\(67\!\cdots\!24\)\( p^{5} T^{5} + \)\(16\!\cdots\!16\)\( p^{6} T^{6} - \)\(67\!\cdots\!24\)\( p^{74} T^{7} + \)\(57\!\cdots\!55\)\( p^{142} T^{8} - \)\(16\!\cdots\!40\)\( p^{210} T^{9} + \)\(11\!\cdots\!78\)\( p^{278} T^{10} - \)\(19\!\cdots\!44\)\( p^{346} T^{11} + p^{414} T^{12} \) | |
| 71 | \( 1 + \)\(80\!\cdots\!68\)\( T + \)\(22\!\cdots\!46\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!08\)\( T^{5} + \)\(14\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!08\)\( p^{69} T^{7} + \)\(22\!\cdots\!95\)\( p^{138} T^{8} + \)\(15\!\cdots\!80\)\( p^{207} T^{9} + \)\(22\!\cdots\!46\)\( p^{276} T^{10} + \)\(80\!\cdots\!68\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 73 | \( 1 - \)\(23\!\cdots\!12\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(63\!\cdots\!91\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(30\!\cdots\!12\)\( T^{6} - \)\(10\!\cdots\!16\)\( p^{69} T^{7} + \)\(63\!\cdots\!91\)\( p^{138} T^{8} - \)\(16\!\cdots\!04\)\( p^{207} T^{9} + \)\(10\!\cdots\!46\)\( p^{276} T^{10} - \)\(23\!\cdots\!12\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 79 | \( 1 - \)\(15\!\cdots\!40\)\( T + \)\(15\!\cdots\!14\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!15\)\( T^{4} + \)\(32\!\cdots\!00\)\( T^{5} + \)\(63\!\cdots\!80\)\( T^{6} + \)\(32\!\cdots\!00\)\( p^{69} T^{7} + \)\(81\!\cdots\!15\)\( p^{138} T^{8} + \)\(94\!\cdots\!00\)\( p^{207} T^{9} + \)\(15\!\cdots\!14\)\( p^{276} T^{10} - \)\(15\!\cdots\!40\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 83 | \( 1 + \)\(21\!\cdots\!04\)\( T + \)\(15\!\cdots\!26\)\( T^{2} + \)\(26\!\cdots\!88\)\( T^{3} + \)\(99\!\cdots\!27\)\( T^{4} + \)\(13\!\cdots\!88\)\( T^{5} + \)\(34\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!88\)\( p^{69} T^{7} + \)\(99\!\cdots\!27\)\( p^{138} T^{8} + \)\(26\!\cdots\!88\)\( p^{207} T^{9} + \)\(15\!\cdots\!26\)\( p^{276} T^{10} + \)\(21\!\cdots\!04\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 89 | \( 1 + \)\(66\!\cdots\!28\)\( T + \)\(74\!\cdots\!42\)\( T^{2} + \)\(50\!\cdots\!56\)\( T^{3} + \)\(46\!\cdots\!43\)\( T^{4} + \)\(25\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} + \)\(25\!\cdots\!72\)\( p^{69} T^{7} + \)\(46\!\cdots\!43\)\( p^{138} T^{8} + \)\(50\!\cdots\!56\)\( p^{207} T^{9} + \)\(74\!\cdots\!42\)\( p^{276} T^{10} + \)\(66\!\cdots\!28\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
| 97 | \( 1 + \)\(11\!\cdots\!32\)\( T + \)\(92\!\cdots\!62\)\( T^{2} + \)\(54\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!55\)\( T^{4} + \)\(11\!\cdots\!32\)\( T^{5} + \)\(43\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!32\)\( p^{69} T^{7} + \)\(27\!\cdots\!55\)\( p^{138} T^{8} + \)\(54\!\cdots\!80\)\( p^{207} T^{9} + \)\(92\!\cdots\!62\)\( p^{276} T^{10} + \)\(11\!\cdots\!32\)\( p^{345} T^{11} + p^{414} T^{12} \) | |
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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
−6.46499429518664764051111609678, −6.40177404014308544151475642908, −5.62290442526079231187906511866, −5.47255502142071871931370878808, −5.45623551851736140101199562867, −5.43145751222861927776160800325, −5.01953312161666652267425467783, −4.72870129633009344222994879798, −4.65517760829454872135677224867, −4.40065807158225104821366751367, −4.21875962731481771910258465114, −4.10041574669606384290306437639, −4.09086203209559122514600203223, −3.31241236444093767663255220474, −3.17390441139435861776994405403, −3.00727083282483829269984404513, −2.63117137389992124847785363945, −2.26422138083663396253397276184, −2.16545218842423932866201896474, −1.99622477141670619207575664603, −1.38039101775519161690780486305, −1.36291062010476050067988811176, −1.10889030032781989446526391078, −1.01359633001550652391109754128, −0.64563869738729043797498623460, 0, 0, 0, 0, 0, 0, 0.64563869738729043797498623460, 1.01359633001550652391109754128, 1.10889030032781989446526391078, 1.36291062010476050067988811176, 1.38039101775519161690780486305, 1.99622477141670619207575664603, 2.16545218842423932866201896474, 2.26422138083663396253397276184, 2.63117137389992124847785363945, 3.00727083282483829269984404513, 3.17390441139435861776994405403, 3.31241236444093767663255220474, 4.09086203209559122514600203223, 4.10041574669606384290306437639, 4.21875962731481771910258465114, 4.40065807158225104821366751367, 4.65517760829454872135677224867, 4.72870129633009344222994879798, 5.01953312161666652267425467783, 5.43145751222861927776160800325, 5.45623551851736140101199562867, 5.47255502142071871931370878808, 5.62290442526079231187906511866, 6.40177404014308544151475642908, 6.46499429518664764051111609678
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.