Dirichlet series
| L(s) = 1 | + 7.29e10·2-s + 7.30e20·4-s − 1.01e25·5-s + 5.99e29·7-s − 1.10e31·8-s − 7.36e35·10-s − 2.21e37·11-s − 3.73e39·13-s + 4.37e40·14-s + 2.03e42·16-s − 2.97e43·17-s − 3.40e45·19-s − 7.38e45·20-s − 1.61e48·22-s + 2.41e47·23-s − 2.34e48·25-s − 2.72e50·26-s + 4.38e50·28-s − 1.40e52·29-s − 2.70e53·31-s + 8.35e52·32-s − 2.16e54·34-s − 6.05e54·35-s − 3.57e55·37-s − 2.47e56·38-s + 1.12e56·40-s − 4.77e56·41-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.309·4-s − 1.55·5-s + 0.598·7-s − 0.0966·8-s − 2.33·10-s − 2.37·11-s − 1.06·13-s + 0.897·14-s + 0.364·16-s − 0.619·17-s − 1.36·19-s − 0.480·20-s − 3.56·22-s + 0.109·23-s − 0.0554·25-s − 1.59·26-s + 0.185·28-s − 1.70·29-s − 3.08·31-s + 0.308·32-s − 0.929·34-s − 0.929·35-s − 0.763·37-s − 2.05·38-s + 0.150·40-s − 0.266·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+71/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(531441\) = \(3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.62612\times 10^{14}\) |
| Root analytic conductor: | \(16.9505\) |
| Motivic weight: | \(71\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 531441,\ (\ :[71/2]^{6}),\ 1)\) |
Particular Values
| \(L(36)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{73}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 36451828413 p T + 35812641475745460897 p^{7} T^{2} - \)\(41\!\cdots\!57\)\( p^{16} T^{3} + \)\(12\!\cdots\!53\)\( p^{30} T^{4} - \)\(23\!\cdots\!77\)\( p^{48} T^{5} + \)\(10\!\cdots\!41\)\( p^{68} T^{6} - \)\(23\!\cdots\!77\)\( p^{119} T^{7} + \)\(12\!\cdots\!53\)\( p^{172} T^{8} - \)\(41\!\cdots\!57\)\( p^{229} T^{9} + 35812641475745460897 p^{291} T^{10} - 36451828413 p^{356} T^{11} + p^{426} T^{12} \) |
| 5 | \( 1 + \)\(40\!\cdots\!96\)\( p^{2} T + \)\(66\!\cdots\!66\)\( p^{6} T^{2} + \)\(34\!\cdots\!56\)\( p^{11} T^{3} - \)\(76\!\cdots\!33\)\( p^{17} T^{4} - \)\(10\!\cdots\!12\)\( p^{25} T^{5} - \)\(11\!\cdots\!32\)\( p^{36} T^{6} - \)\(10\!\cdots\!12\)\( p^{96} T^{7} - \)\(76\!\cdots\!33\)\( p^{159} T^{8} + \)\(34\!\cdots\!56\)\( p^{224} T^{9} + \)\(66\!\cdots\!66\)\( p^{290} T^{10} + \)\(40\!\cdots\!96\)\( p^{357} T^{11} + p^{426} T^{12} \) | |
| 7 | \( 1 - \)\(59\!\cdots\!88\)\( T + \)\(72\!\cdots\!54\)\( p^{2} T^{2} - \)\(17\!\cdots\!36\)\( p^{4} T^{3} + \)\(17\!\cdots\!39\)\( p^{10} T^{4} + \)\(45\!\cdots\!16\)\( p^{16} T^{5} + \)\(24\!\cdots\!52\)\( p^{24} T^{6} + \)\(45\!\cdots\!16\)\( p^{87} T^{7} + \)\(17\!\cdots\!39\)\( p^{152} T^{8} - \)\(17\!\cdots\!36\)\( p^{217} T^{9} + \)\(72\!\cdots\!54\)\( p^{286} T^{10} - \)\(59\!\cdots\!88\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 11 | \( 1 + \)\(22\!\cdots\!64\)\( T + \)\(47\!\cdots\!14\)\( p T^{2} + \)\(62\!\cdots\!08\)\( p^{3} T^{3} + \)\(58\!\cdots\!53\)\( p^{7} T^{4} + \)\(43\!\cdots\!20\)\( p^{12} T^{5} + \)\(26\!\cdots\!28\)\( p^{17} T^{6} + \)\(43\!\cdots\!20\)\( p^{83} T^{7} + \)\(58\!\cdots\!53\)\( p^{149} T^{8} + \)\(62\!\cdots\!08\)\( p^{216} T^{9} + \)\(47\!\cdots\!14\)\( p^{285} T^{10} + \)\(22\!\cdots\!64\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 13 | \( 1 + \)\(37\!\cdots\!76\)\( T + \)\(34\!\cdots\!30\)\( p T^{2} + \)\(62\!\cdots\!56\)\( p^{3} T^{3} + \)\(21\!\cdots\!95\)\( p^{6} T^{4} + \)\(20\!\cdots\!04\)\( p^{10} T^{5} + \)\(31\!\cdots\!92\)\( p^{15} T^{6} + \)\(20\!\cdots\!04\)\( p^{81} T^{7} + \)\(21\!\cdots\!95\)\( p^{148} T^{8} + \)\(62\!\cdots\!56\)\( p^{216} T^{9} + \)\(34\!\cdots\!30\)\( p^{285} T^{10} + \)\(37\!\cdots\!76\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 17 | \( 1 + \)\(29\!\cdots\!96\)\( T + \)\(29\!\cdots\!42\)\( p^{2} T^{2} + \)\(16\!\cdots\!60\)\( p^{4} T^{3} + \)\(85\!\cdots\!35\)\( p^{7} T^{4} + \)\(19\!\cdots\!24\)\( p^{10} T^{5} + \)\(97\!\cdots\!88\)\( p^{13} T^{6} + \)\(19\!\cdots\!24\)\( p^{81} T^{7} + \)\(85\!\cdots\!35\)\( p^{149} T^{8} + \)\(16\!\cdots\!60\)\( p^{217} T^{9} + \)\(29\!\cdots\!42\)\( p^{286} T^{10} + \)\(29\!\cdots\!96\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 19 | \( 1 + \)\(34\!\cdots\!72\)\( T + \)\(17\!\cdots\!82\)\( T^{2} + \)\(25\!\cdots\!36\)\( p T^{3} + \)\(28\!\cdots\!97\)\( p^{3} T^{4} + \)\(16\!\cdots\!72\)\( p^{5} T^{5} + \)\(76\!\cdots\!68\)\( p^{8} T^{6} + \)\(16\!\cdots\!72\)\( p^{76} T^{7} + \)\(28\!\cdots\!97\)\( p^{145} T^{8} + \)\(25\!\cdots\!36\)\( p^{214} T^{9} + \)\(17\!\cdots\!82\)\( p^{284} T^{10} + \)\(34\!\cdots\!72\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 23 | \( 1 - \)\(10\!\cdots\!88\)\( p T + \)\(39\!\cdots\!06\)\( p^{2} T^{2} - \)\(31\!\cdots\!92\)\( p^{4} T^{3} + \)\(12\!\cdots\!99\)\( p^{6} T^{4} - \)\(12\!\cdots\!92\)\( p^{8} T^{5} + \)\(26\!\cdots\!32\)\( p^{10} T^{6} - \)\(12\!\cdots\!92\)\( p^{79} T^{7} + \)\(12\!\cdots\!99\)\( p^{148} T^{8} - \)\(31\!\cdots\!92\)\( p^{217} T^{9} + \)\(39\!\cdots\!06\)\( p^{286} T^{10} - \)\(10\!\cdots\!88\)\( p^{356} T^{11} + p^{426} T^{12} \) | |
| 29 | \( 1 + \)\(14\!\cdots\!96\)\( T + \)\(13\!\cdots\!74\)\( p T^{2} + \)\(46\!\cdots\!68\)\( p^{2} T^{3} + \)\(85\!\cdots\!07\)\( p^{4} T^{4} + \)\(80\!\cdots\!56\)\( p^{6} T^{5} + \)\(10\!\cdots\!72\)\( p^{8} T^{6} + \)\(80\!\cdots\!56\)\( p^{77} T^{7} + \)\(85\!\cdots\!07\)\( p^{146} T^{8} + \)\(46\!\cdots\!68\)\( p^{215} T^{9} + \)\(13\!\cdots\!74\)\( p^{285} T^{10} + \)\(14\!\cdots\!96\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 31 | \( 1 + \)\(27\!\cdots\!24\)\( T + \)\(19\!\cdots\!38\)\( p T^{2} + \)\(94\!\cdots\!12\)\( p^{2} T^{3} + \)\(40\!\cdots\!53\)\( p^{3} T^{4} + \)\(44\!\cdots\!56\)\( p^{5} T^{5} + \)\(45\!\cdots\!72\)\( p^{7} T^{6} + \)\(44\!\cdots\!56\)\( p^{76} T^{7} + \)\(40\!\cdots\!53\)\( p^{145} T^{8} + \)\(94\!\cdots\!12\)\( p^{215} T^{9} + \)\(19\!\cdots\!38\)\( p^{285} T^{10} + \)\(27\!\cdots\!24\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 37 | \( 1 + \)\(35\!\cdots\!52\)\( T + \)\(80\!\cdots\!06\)\( T^{2} + \)\(14\!\cdots\!64\)\( T^{3} + \)\(70\!\cdots\!23\)\( p T^{4} + \)\(10\!\cdots\!24\)\( p^{2} T^{5} + \)\(11\!\cdots\!64\)\( p^{3} T^{6} + \)\(10\!\cdots\!24\)\( p^{73} T^{7} + \)\(70\!\cdots\!23\)\( p^{143} T^{8} + \)\(14\!\cdots\!64\)\( p^{213} T^{9} + \)\(80\!\cdots\!06\)\( p^{284} T^{10} + \)\(35\!\cdots\!52\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 41 | \( 1 + \)\(47\!\cdots\!16\)\( T + \)\(86\!\cdots\!78\)\( T^{2} + \)\(22\!\cdots\!28\)\( p T^{3} + \)\(29\!\cdots\!43\)\( p^{2} T^{4} + \)\(67\!\cdots\!44\)\( p^{3} T^{5} + \)\(70\!\cdots\!92\)\( p^{4} T^{6} + \)\(67\!\cdots\!44\)\( p^{74} T^{7} + \)\(29\!\cdots\!43\)\( p^{144} T^{8} + \)\(22\!\cdots\!28\)\( p^{214} T^{9} + \)\(86\!\cdots\!78\)\( p^{284} T^{10} + \)\(47\!\cdots\!16\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 43 | \( 1 - \)\(11\!\cdots\!80\)\( T + \)\(30\!\cdots\!90\)\( T^{2} - \)\(23\!\cdots\!60\)\( T^{3} + \)\(91\!\cdots\!29\)\( p T^{4} - \)\(11\!\cdots\!60\)\( p^{2} T^{5} + \)\(45\!\cdots\!60\)\( p^{3} T^{6} - \)\(11\!\cdots\!60\)\( p^{73} T^{7} + \)\(91\!\cdots\!29\)\( p^{143} T^{8} - \)\(23\!\cdots\!60\)\( p^{213} T^{9} + \)\(30\!\cdots\!90\)\( p^{284} T^{10} - \)\(11\!\cdots\!80\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 47 | \( 1 - \)\(74\!\cdots\!60\)\( T + \)\(23\!\cdots\!70\)\( T^{2} - \)\(42\!\cdots\!40\)\( p T^{3} + \)\(11\!\cdots\!03\)\( p^{2} T^{4} - \)\(18\!\cdots\!60\)\( p^{3} T^{5} + \)\(35\!\cdots\!60\)\( p^{4} T^{6} - \)\(18\!\cdots\!60\)\( p^{74} T^{7} + \)\(11\!\cdots\!03\)\( p^{144} T^{8} - \)\(42\!\cdots\!40\)\( p^{214} T^{9} + \)\(23\!\cdots\!70\)\( p^{284} T^{10} - \)\(74\!\cdots\!60\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 53 | \( 1 + \)\(54\!\cdots\!16\)\( T + \)\(33\!\cdots\!78\)\( p T^{2} + \)\(15\!\cdots\!12\)\( p^{2} T^{3} + \)\(59\!\cdots\!43\)\( p^{3} T^{4} + \)\(19\!\cdots\!68\)\( p^{4} T^{5} + \)\(61\!\cdots\!76\)\( p^{5} T^{6} + \)\(19\!\cdots\!68\)\( p^{75} T^{7} + \)\(59\!\cdots\!43\)\( p^{145} T^{8} + \)\(15\!\cdots\!12\)\( p^{215} T^{9} + \)\(33\!\cdots\!78\)\( p^{285} T^{10} + \)\(54\!\cdots\!16\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 59 | \( 1 - \)\(29\!\cdots\!48\)\( T + \)\(98\!\cdots\!18\)\( p T^{2} - \)\(23\!\cdots\!56\)\( p^{2} T^{3} + \)\(46\!\cdots\!17\)\( p^{3} T^{4} - \)\(75\!\cdots\!12\)\( p^{4} T^{5} + \)\(10\!\cdots\!12\)\( p^{5} T^{6} - \)\(75\!\cdots\!12\)\( p^{75} T^{7} + \)\(46\!\cdots\!17\)\( p^{145} T^{8} - \)\(23\!\cdots\!56\)\( p^{215} T^{9} + \)\(98\!\cdots\!18\)\( p^{285} T^{10} - \)\(29\!\cdots\!48\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 61 | \( 1 - \)\(20\!\cdots\!20\)\( T + \)\(45\!\cdots\!74\)\( p T^{2} - \)\(24\!\cdots\!40\)\( p^{3} T^{3} + \)\(15\!\cdots\!95\)\( p^{3} T^{4} - \)\(45\!\cdots\!40\)\( p^{4} T^{5} + \)\(30\!\cdots\!60\)\( p^{5} T^{6} - \)\(45\!\cdots\!40\)\( p^{75} T^{7} + \)\(15\!\cdots\!95\)\( p^{145} T^{8} - \)\(24\!\cdots\!40\)\( p^{216} T^{9} + \)\(45\!\cdots\!74\)\( p^{285} T^{10} - \)\(20\!\cdots\!20\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 67 | \( 1 + \)\(35\!\cdots\!84\)\( T + \)\(24\!\cdots\!14\)\( p T^{2} + \)\(14\!\cdots\!60\)\( p^{2} T^{3} + \)\(41\!\cdots\!85\)\( p^{3} T^{4} + \)\(26\!\cdots\!44\)\( p^{4} T^{5} + \)\(46\!\cdots\!68\)\( p^{5} T^{6} + \)\(26\!\cdots\!44\)\( p^{75} T^{7} + \)\(41\!\cdots\!85\)\( p^{145} T^{8} + \)\(14\!\cdots\!60\)\( p^{215} T^{9} + \)\(24\!\cdots\!14\)\( p^{285} T^{10} + \)\(35\!\cdots\!84\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 71 | \( 1 - \)\(51\!\cdots\!28\)\( T + \)\(36\!\cdots\!86\)\( T^{2} - \)\(67\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(54\!\cdots\!08\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} - \)\(54\!\cdots\!08\)\( p^{71} T^{7} + \)\(11\!\cdots\!95\)\( p^{142} T^{8} - \)\(67\!\cdots\!80\)\( p^{213} T^{9} + \)\(36\!\cdots\!86\)\( p^{284} T^{10} - \)\(51\!\cdots\!28\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 73 | \( 1 + \)\(19\!\cdots\!24\)\( T + \)\(50\!\cdots\!34\)\( T^{2} + \)\(76\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!47\)\( p T^{4} + \)\(24\!\cdots\!12\)\( T^{5} + \)\(41\!\cdots\!68\)\( T^{6} + \)\(24\!\cdots\!12\)\( p^{71} T^{7} + \)\(24\!\cdots\!47\)\( p^{143} T^{8} + \)\(76\!\cdots\!32\)\( p^{213} T^{9} + \)\(50\!\cdots\!34\)\( p^{284} T^{10} + \)\(19\!\cdots\!24\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 79 | \( 1 - \)\(48\!\cdots\!60\)\( T + \)\(39\!\cdots\!74\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(44\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!00\)\( p^{71} T^{7} + \)\(59\!\cdots\!15\)\( p^{142} T^{8} - \)\(13\!\cdots\!00\)\( p^{213} T^{9} + \)\(39\!\cdots\!74\)\( p^{284} T^{10} - \)\(48\!\cdots\!60\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 83 | \( 1 + \)\(23\!\cdots\!68\)\( T + \)\(83\!\cdots\!14\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!67\)\( T^{4} + \)\(41\!\cdots\!16\)\( T^{5} + \)\(70\!\cdots\!44\)\( T^{6} + \)\(41\!\cdots\!16\)\( p^{71} T^{7} + \)\(30\!\cdots\!67\)\( p^{142} T^{8} + \)\(13\!\cdots\!44\)\( p^{213} T^{9} + \)\(83\!\cdots\!14\)\( p^{284} T^{10} + \)\(23\!\cdots\!68\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 89 | \( 1 + \)\(43\!\cdots\!08\)\( T + \)\(18\!\cdots\!82\)\( T^{2} + \)\(46\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!63\)\( T^{4} + \)\(21\!\cdots\!72\)\( T^{5} + \)\(37\!\cdots\!08\)\( T^{6} + \)\(21\!\cdots\!72\)\( p^{71} T^{7} + \)\(11\!\cdots\!63\)\( p^{142} T^{8} + \)\(46\!\cdots\!36\)\( p^{213} T^{9} + \)\(18\!\cdots\!82\)\( p^{284} T^{10} + \)\(43\!\cdots\!08\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
| 97 | \( 1 - \)\(97\!\cdots\!96\)\( T + \)\(10\!\cdots\!58\)\( T^{2} - \)\(61\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!76\)\( T^{5} + \)\(58\!\cdots\!56\)\( T^{6} - \)\(14\!\cdots\!76\)\( p^{71} T^{7} + \)\(36\!\cdots\!55\)\( p^{142} T^{8} - \)\(61\!\cdots\!60\)\( p^{213} T^{9} + \)\(10\!\cdots\!58\)\( p^{284} T^{10} - \)\(97\!\cdots\!96\)\( p^{355} T^{11} + p^{426} T^{12} \) | |
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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
−5.11788175901657633734245578913, −4.87499277480833762154902742541, −4.65169180885172124123774594032, −4.38906975429583804327154339200, −4.28198679929607729785242181218, −4.27819097193620167398665306164, −4.23151672782841111410174511226, −3.81016144590784535080723945371, −3.61925345096503100427148864700, −3.42146891264675200577300635554, −3.37544537224278813371879746437, −3.26852655318156501811289799868, −2.99188762066438499462619904977, −2.63291785882616383235542608040, −2.56762767775414945136647072516, −2.35274722657591881079063529065, −2.08899888416576824435210236606, −1.96543458061019589596312779114, −1.84042073011377192811123934547, −1.79374942934747996956321327927, −1.58423762054456403089187721425, −1.16854490470271887205309622751, −0.951793704272535803111443817681, −0.825822172610631809014135049171, −0.59639476492165348413975509180, 0, 0, 0, 0, 0, 0, 0.59639476492165348413975509180, 0.825822172610631809014135049171, 0.951793704272535803111443817681, 1.16854490470271887205309622751, 1.58423762054456403089187721425, 1.79374942934747996956321327927, 1.84042073011377192811123934547, 1.96543458061019589596312779114, 2.08899888416576824435210236606, 2.35274722657591881079063529065, 2.56762767775414945136647072516, 2.63291785882616383235542608040, 2.99188762066438499462619904977, 3.26852655318156501811289799868, 3.37544537224278813371879746437, 3.42146891264675200577300635554, 3.61925345096503100427148864700, 3.81016144590784535080723945371, 4.23151672782841111410174511226, 4.27819097193620167398665306164, 4.28198679929607729785242181218, 4.38906975429583804327154339200, 4.65169180885172124123774594032, 4.87499277480833762154902742541, 5.11788175901657633734245578913
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.