Dirichlet series
| L(s) = 1 | − 1.37e10·2-s − 1.17e20·4-s + 1.80e23·5-s − 2.80e28·7-s + 3.09e30·8-s − 2.47e33·10-s − 1.25e33·11-s − 1.03e36·13-s + 3.85e38·14-s − 2.42e40·16-s + 2.23e40·17-s − 6.70e42·19-s − 2.12e43·20-s + 1.72e43·22-s − 1.22e46·23-s − 1.24e47·25-s + 1.41e46·26-s + 3.30e48·28-s + 1.88e49·29-s − 2.24e50·31-s + 1.11e50·32-s − 3.06e50·34-s − 5.06e51·35-s + 2.74e52·37-s + 9.21e52·38-s + 5.58e53·40-s − 2.66e54·41-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.798·4-s + 0.692·5-s − 1.37·7-s + 1.72·8-s − 0.783·10-s − 0.0162·11-s − 0.0497·13-s + 1.55·14-s − 1.11·16-s + 0.134·17-s − 0.973·19-s − 0.552·20-s + 0.0184·22-s − 2.95·23-s − 1.83·25-s + 0.0562·26-s + 1.09·28-s + 1.92·29-s − 2.45·31-s + 0.419·32-s − 0.151·34-s − 0.950·35-s + 0.800·37-s + 1.10·38-s + 1.19·40-s − 2.49·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+67/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(531441\) = \(3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.80561\times 10^{14}\) |
| Root analytic conductor: | \(15.9956\) |
| Motivic weight: | \(67\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 531441,\ (\ :[67/2]^{6}),\ 1)\) |
Particular Values
| \(L(34)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{69}{2})\) | 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 | \( 1 + 6867677583 p T + 4788799754343454299 p^{6} T^{2} + \)\(83\!\cdots\!19\)\( p^{15} T^{3} + \)\(51\!\cdots\!53\)\( p^{30} T^{4} + \)\(24\!\cdots\!09\)\( p^{47} T^{5} + \)\(89\!\cdots\!99\)\( p^{66} T^{6} + \)\(24\!\cdots\!09\)\( p^{114} T^{7} + \)\(51\!\cdots\!53\)\( p^{164} T^{8} + \)\(83\!\cdots\!19\)\( p^{216} T^{9} + 4788799754343454299 p^{274} T^{10} + 6867677583 p^{336} T^{11} + p^{402} T^{12} \) |
| 5 | \( 1 - \)\(72\!\cdots\!92\)\( p^{2} T + \)\(50\!\cdots\!78\)\( p^{5} T^{2} - \)\(14\!\cdots\!88\)\( p^{9} T^{3} + \)\(17\!\cdots\!91\)\( p^{17} T^{4} - \)\(24\!\cdots\!48\)\( p^{27} T^{5} + \)\(24\!\cdots\!12\)\( p^{38} T^{6} - \)\(24\!\cdots\!48\)\( p^{94} T^{7} + \)\(17\!\cdots\!91\)\( p^{151} T^{8} - \)\(14\!\cdots\!88\)\( p^{210} T^{9} + \)\(50\!\cdots\!78\)\( p^{273} T^{10} - \)\(72\!\cdots\!92\)\( p^{337} T^{11} + p^{402} T^{12} \) | |
| 7 | \( 1 + \)\(28\!\cdots\!08\)\( T + \)\(16\!\cdots\!18\)\( p T^{2} + \)\(55\!\cdots\!36\)\( p^{4} T^{3} + \)\(91\!\cdots\!11\)\( p^{8} T^{4} + \)\(96\!\cdots\!16\)\( p^{14} T^{5} + \)\(51\!\cdots\!16\)\( p^{21} T^{6} + \)\(96\!\cdots\!16\)\( p^{81} T^{7} + \)\(91\!\cdots\!11\)\( p^{142} T^{8} + \)\(55\!\cdots\!36\)\( p^{205} T^{9} + \)\(16\!\cdots\!18\)\( p^{269} T^{10} + \)\(28\!\cdots\!08\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 11 | \( 1 + \)\(12\!\cdots\!24\)\( T + \)\(21\!\cdots\!54\)\( p^{2} T^{2} - \)\(48\!\cdots\!12\)\( p^{5} T^{3} + \)\(14\!\cdots\!23\)\( p^{8} T^{4} - \)\(52\!\cdots\!60\)\( p^{11} T^{5} + \)\(57\!\cdots\!28\)\( p^{15} T^{6} - \)\(52\!\cdots\!60\)\( p^{78} T^{7} + \)\(14\!\cdots\!23\)\( p^{142} T^{8} - \)\(48\!\cdots\!12\)\( p^{206} T^{9} + \)\(21\!\cdots\!54\)\( p^{270} T^{10} + \)\(12\!\cdots\!24\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 13 | \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(46\!\cdots\!70\)\( p T^{2} - \)\(52\!\cdots\!96\)\( p^{3} T^{3} + \)\(56\!\cdots\!35\)\( p^{6} T^{4} - \)\(81\!\cdots\!52\)\( p^{9} T^{5} + \)\(44\!\cdots\!24\)\( p^{12} T^{6} - \)\(81\!\cdots\!52\)\( p^{76} T^{7} + \)\(56\!\cdots\!35\)\( p^{140} T^{8} - \)\(52\!\cdots\!96\)\( p^{204} T^{9} + \)\(46\!\cdots\!70\)\( p^{269} T^{10} + \)\(10\!\cdots\!64\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 17 | \( 1 - \)\(13\!\cdots\!88\)\( p T + \)\(26\!\cdots\!02\)\( p^{2} T^{2} + \)\(65\!\cdots\!80\)\( p^{3} T^{3} + \)\(31\!\cdots\!55\)\( p^{4} T^{4} + \)\(72\!\cdots\!88\)\( p^{7} T^{5} + \)\(34\!\cdots\!44\)\( p^{10} T^{6} + \)\(72\!\cdots\!88\)\( p^{74} T^{7} + \)\(31\!\cdots\!55\)\( p^{138} T^{8} + \)\(65\!\cdots\!80\)\( p^{204} T^{9} + \)\(26\!\cdots\!02\)\( p^{270} T^{10} - \)\(13\!\cdots\!88\)\( p^{336} T^{11} + p^{402} T^{12} \) | |
| 19 | \( 1 + \)\(67\!\cdots\!52\)\( T + \)\(82\!\cdots\!42\)\( T^{2} + \)\(20\!\cdots\!56\)\( p T^{3} + \)\(17\!\cdots\!17\)\( p^{3} T^{4} - \)\(28\!\cdots\!12\)\( p^{6} T^{5} - \)\(41\!\cdots\!12\)\( p^{8} T^{6} - \)\(28\!\cdots\!12\)\( p^{73} T^{7} + \)\(17\!\cdots\!17\)\( p^{137} T^{8} + \)\(20\!\cdots\!56\)\( p^{202} T^{9} + \)\(82\!\cdots\!42\)\( p^{268} T^{10} + \)\(67\!\cdots\!52\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 23 | \( 1 + \)\(12\!\cdots\!64\)\( T + \)\(91\!\cdots\!54\)\( T^{2} + \)\(17\!\cdots\!24\)\( p T^{3} + \)\(19\!\cdots\!39\)\( p^{2} T^{4} - \)\(15\!\cdots\!28\)\( p^{4} T^{5} - \)\(62\!\cdots\!08\)\( p^{6} T^{6} - \)\(15\!\cdots\!28\)\( p^{71} T^{7} + \)\(19\!\cdots\!39\)\( p^{136} T^{8} + \)\(17\!\cdots\!24\)\( p^{202} T^{9} + \)\(91\!\cdots\!54\)\( p^{268} T^{10} + \)\(12\!\cdots\!64\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 29 | \( 1 - \)\(18\!\cdots\!64\)\( T + \)\(17\!\cdots\!54\)\( p T^{2} - \)\(73\!\cdots\!52\)\( p^{2} T^{3} + \)\(13\!\cdots\!07\)\( p^{4} T^{4} - \)\(15\!\cdots\!64\)\( p^{6} T^{5} + \)\(21\!\cdots\!52\)\( p^{8} T^{6} - \)\(15\!\cdots\!64\)\( p^{73} T^{7} + \)\(13\!\cdots\!07\)\( p^{138} T^{8} - \)\(73\!\cdots\!52\)\( p^{203} T^{9} + \)\(17\!\cdots\!54\)\( p^{269} T^{10} - \)\(18\!\cdots\!64\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 31 | \( 1 + \)\(22\!\cdots\!04\)\( T + \)\(21\!\cdots\!98\)\( p T^{2} + \)\(98\!\cdots\!32\)\( p^{2} T^{3} + \)\(51\!\cdots\!13\)\( p^{3} T^{4} + \)\(17\!\cdots\!16\)\( p^{6} T^{5} + \)\(63\!\cdots\!72\)\( p^{7} T^{6} + \)\(17\!\cdots\!16\)\( p^{73} T^{7} + \)\(51\!\cdots\!13\)\( p^{137} T^{8} + \)\(98\!\cdots\!32\)\( p^{203} T^{9} + \)\(21\!\cdots\!98\)\( p^{269} T^{10} + \)\(22\!\cdots\!04\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 37 | \( 1 - \)\(27\!\cdots\!52\)\( T + \)\(27\!\cdots\!26\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(77\!\cdots\!43\)\( p T^{4} + \)\(58\!\cdots\!16\)\( p^{2} T^{5} + \)\(33\!\cdots\!84\)\( p^{3} T^{6} + \)\(58\!\cdots\!16\)\( p^{69} T^{7} + \)\(77\!\cdots\!43\)\( p^{135} T^{8} - \)\(10\!\cdots\!64\)\( p^{201} T^{9} + \)\(27\!\cdots\!26\)\( p^{268} T^{10} - \)\(27\!\cdots\!52\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 41 | \( 1 + \)\(26\!\cdots\!96\)\( T + \)\(71\!\cdots\!58\)\( T^{2} + \)\(30\!\cdots\!28\)\( p T^{3} + \)\(12\!\cdots\!03\)\( p^{2} T^{4} + \)\(93\!\cdots\!44\)\( p^{4} T^{5} + \)\(11\!\cdots\!52\)\( p^{4} T^{6} + \)\(93\!\cdots\!44\)\( p^{71} T^{7} + \)\(12\!\cdots\!03\)\( p^{136} T^{8} + \)\(30\!\cdots\!28\)\( p^{202} T^{9} + \)\(71\!\cdots\!58\)\( p^{268} T^{10} + \)\(26\!\cdots\!96\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 43 | \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(98\!\cdots\!10\)\( T^{2} + \)\(79\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!29\)\( p T^{4} + \)\(45\!\cdots\!00\)\( p^{3} T^{5} + \)\(22\!\cdots\!40\)\( p^{3} T^{6} + \)\(45\!\cdots\!00\)\( p^{70} T^{7} + \)\(12\!\cdots\!29\)\( p^{135} T^{8} + \)\(79\!\cdots\!00\)\( p^{201} T^{9} + \)\(98\!\cdots\!10\)\( p^{268} T^{10} + \)\(11\!\cdots\!00\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 47 | \( 1 + \)\(58\!\cdots\!60\)\( T + \)\(21\!\cdots\!70\)\( T^{2} + \)\(53\!\cdots\!40\)\( p T^{3} + \)\(15\!\cdots\!23\)\( p^{2} T^{4} - \)\(33\!\cdots\!40\)\( p^{3} T^{5} + \)\(76\!\cdots\!60\)\( p^{4} T^{6} - \)\(33\!\cdots\!40\)\( p^{70} T^{7} + \)\(15\!\cdots\!23\)\( p^{136} T^{8} + \)\(53\!\cdots\!40\)\( p^{202} T^{9} + \)\(21\!\cdots\!70\)\( p^{268} T^{10} + \)\(58\!\cdots\!60\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 53 | \( 1 - \)\(13\!\cdots\!36\)\( T + \)\(40\!\cdots\!58\)\( p T^{2} - \)\(67\!\cdots\!52\)\( p^{2} T^{3} + \)\(11\!\cdots\!43\)\( p^{3} T^{4} - \)\(14\!\cdots\!88\)\( p^{4} T^{5} + \)\(18\!\cdots\!56\)\( p^{5} T^{6} - \)\(14\!\cdots\!88\)\( p^{71} T^{7} + \)\(11\!\cdots\!43\)\( p^{137} T^{8} - \)\(67\!\cdots\!52\)\( p^{203} T^{9} + \)\(40\!\cdots\!58\)\( p^{269} T^{10} - \)\(13\!\cdots\!36\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 59 | \( 1 + \)\(43\!\cdots\!12\)\( T + \)\(38\!\cdots\!18\)\( p T^{2} + \)\(17\!\cdots\!84\)\( p^{2} T^{3} + \)\(10\!\cdots\!97\)\( p^{3} T^{4} + \)\(36\!\cdots\!88\)\( p^{4} T^{5} + \)\(15\!\cdots\!52\)\( p^{5} T^{6} + \)\(36\!\cdots\!88\)\( p^{71} T^{7} + \)\(10\!\cdots\!97\)\( p^{137} T^{8} + \)\(17\!\cdots\!84\)\( p^{203} T^{9} + \)\(38\!\cdots\!18\)\( p^{269} T^{10} + \)\(43\!\cdots\!12\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 61 | \( 1 + \)\(30\!\cdots\!40\)\( T + \)\(19\!\cdots\!34\)\( p T^{2} + \)\(44\!\cdots\!80\)\( p^{2} T^{3} + \)\(30\!\cdots\!15\)\( p^{3} T^{4} + \)\(30\!\cdots\!80\)\( p^{4} T^{5} + \)\(36\!\cdots\!80\)\( p^{5} T^{6} + \)\(30\!\cdots\!80\)\( p^{71} T^{7} + \)\(30\!\cdots\!15\)\( p^{137} T^{8} + \)\(44\!\cdots\!80\)\( p^{203} T^{9} + \)\(19\!\cdots\!34\)\( p^{269} T^{10} + \)\(30\!\cdots\!40\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 67 | \( 1 + \)\(36\!\cdots\!96\)\( T + \)\(83\!\cdots\!78\)\( T^{2} + \)\(16\!\cdots\!80\)\( p T^{3} + \)\(32\!\cdots\!55\)\( T^{4} - \)\(28\!\cdots\!24\)\( T^{5} + \)\(84\!\cdots\!56\)\( T^{6} - \)\(28\!\cdots\!24\)\( p^{67} T^{7} + \)\(32\!\cdots\!55\)\( p^{134} T^{8} + \)\(16\!\cdots\!80\)\( p^{202} T^{9} + \)\(83\!\cdots\!78\)\( p^{268} T^{10} + \)\(36\!\cdots\!96\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 71 | \( 1 + \)\(25\!\cdots\!72\)\( T + \)\(73\!\cdots\!06\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!95\)\( T^{4} + \)\(23\!\cdots\!92\)\( T^{5} + \)\(29\!\cdots\!44\)\( T^{6} + \)\(23\!\cdots\!92\)\( p^{67} T^{7} + \)\(20\!\cdots\!95\)\( p^{134} T^{8} + \)\(11\!\cdots\!20\)\( p^{201} T^{9} + \)\(73\!\cdots\!06\)\( p^{268} T^{10} + \)\(25\!\cdots\!72\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 73 | \( 1 - \)\(16\!\cdots\!24\)\( T + \)\(21\!\cdots\!14\)\( T^{2} - \)\(26\!\cdots\!92\)\( T^{3} + \)\(28\!\cdots\!11\)\( T^{4} - \)\(29\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!28\)\( T^{6} - \)\(29\!\cdots\!92\)\( p^{67} T^{7} + \)\(28\!\cdots\!11\)\( p^{134} T^{8} - \)\(26\!\cdots\!92\)\( p^{201} T^{9} + \)\(21\!\cdots\!14\)\( p^{268} T^{10} - \)\(16\!\cdots\!24\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 79 | \( 1 + \)\(47\!\cdots\!40\)\( T + \)\(66\!\cdots\!54\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!85\)\( p T^{4} + \)\(55\!\cdots\!00\)\( T^{5} + \)\(35\!\cdots\!80\)\( T^{6} + \)\(55\!\cdots\!00\)\( p^{67} T^{7} + \)\(25\!\cdots\!85\)\( p^{135} T^{8} + \)\(23\!\cdots\!00\)\( p^{201} T^{9} + \)\(66\!\cdots\!54\)\( p^{268} T^{10} + \)\(47\!\cdots\!40\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 83 | \( 1 + \)\(36\!\cdots\!32\)\( T + \)\(21\!\cdots\!74\)\( T^{2} + \)\(63\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} + \)\(45\!\cdots\!64\)\( T^{5} + \)\(96\!\cdots\!44\)\( T^{6} + \)\(45\!\cdots\!64\)\( p^{67} T^{7} + \)\(19\!\cdots\!47\)\( p^{134} T^{8} + \)\(63\!\cdots\!56\)\( p^{201} T^{9} + \)\(21\!\cdots\!74\)\( p^{268} T^{10} + \)\(36\!\cdots\!32\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 89 | \( 1 - \)\(24\!\cdots\!92\)\( T + \)\(88\!\cdots\!02\)\( T^{2} + \)\(17\!\cdots\!96\)\( T^{3} + \)\(51\!\cdots\!03\)\( T^{4} - \)\(14\!\cdots\!88\)\( T^{5} + \)\(24\!\cdots\!88\)\( T^{6} - \)\(14\!\cdots\!88\)\( p^{67} T^{7} + \)\(51\!\cdots\!03\)\( p^{134} T^{8} + \)\(17\!\cdots\!96\)\( p^{201} T^{9} + \)\(88\!\cdots\!02\)\( p^{268} T^{10} - \)\(24\!\cdots\!92\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
| 97 | \( 1 - \)\(38\!\cdots\!64\)\( T + \)\(36\!\cdots\!18\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!55\)\( T^{4} - \)\(40\!\cdots\!24\)\( T^{5} + \)\(14\!\cdots\!96\)\( T^{6} - \)\(40\!\cdots\!24\)\( p^{67} T^{7} + \)\(88\!\cdots\!55\)\( p^{134} T^{8} - \)\(19\!\cdots\!40\)\( p^{201} T^{9} + \)\(36\!\cdots\!18\)\( p^{268} T^{10} - \)\(38\!\cdots\!64\)\( p^{335} T^{11} + p^{402} T^{12} \) | |
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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
−5.38129682038449556308566865210, −4.76633314028862890797723185892, −4.61046089882226900514317116674, −4.55868780203918102949253886538, −4.54689180192942620159243850624, −4.40784574261655281128886494993, −4.12290365639855346115663138286, −3.64744753059004879220805457735, −3.63630489585922542999856113290, −3.50834565766096670438145246246, −3.43134000680604443362282777893, −3.30491312264480934736955914493, −2.76554587142690524971762247831, −2.60095744712407385728151388357, −2.56073499697919671911047122850, −2.22068034333520909453353416274, −2.18774765249812397836858145675, −1.93988397292900098097790953724, −1.63404769591927303563949511224, −1.61765155533672895915744309996, −1.46038786584219801078277428704, −1.25900849245042793608010712720, −0.953079819817234846095508402243, −0.804053152321943790634493028603, −0.56587818226941821077479736495, 0, 0, 0, 0, 0, 0, 0.56587818226941821077479736495, 0.804053152321943790634493028603, 0.953079819817234846095508402243, 1.25900849245042793608010712720, 1.46038786584219801078277428704, 1.61765155533672895915744309996, 1.63404769591927303563949511224, 1.93988397292900098097790953724, 2.18774765249812397836858145675, 2.22068034333520909453353416274, 2.56073499697919671911047122850, 2.60095744712407385728151388357, 2.76554587142690524971762247831, 3.30491312264480934736955914493, 3.43134000680604443362282777893, 3.50834565766096670438145246246, 3.63630489585922542999856113290, 3.64744753059004879220805457735, 4.12290365639855346115663138286, 4.40784574261655281128886494993, 4.54689180192942620159243850624, 4.55868780203918102949253886538, 4.61046089882226900514317116674, 4.76633314028862890797723185892, 5.38129682038449556308566865210
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.