Dirichlet series
| L(s) = 1 | + 1.00e8·2-s + 6.00e11·3-s + 5.91e15·4-s + 2.37e17·5-s + 6.04e19·6-s − 1.14e21·7-s + 2.64e23·8-s − 4.12e23·9-s + 2.39e25·10-s + 2.22e25·11-s + 3.54e27·12-s + 2.63e27·13-s − 1.15e29·14-s + 1.42e29·15-s + 9.98e30·16-s − 1.40e30·17-s − 4.15e31·18-s + 1.81e31·19-s + 1.40e33·20-s − 6.90e32·21-s + 2.23e33·22-s + 5.61e33·23-s + 1.58e35·24-s − 2.20e34·25-s + 2.65e35·26-s − 3.26e35·27-s − 6.79e36·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 1.22·3-s + 21/2·4-s + 1.78·5-s + 5.20·6-s − 2.26·7-s + 19.7·8-s − 1.72·9-s + 7.56·10-s + 0.680·11-s + 12.8·12-s + 1.34·13-s − 9.62·14-s + 2.18·15-s + 63/2·16-s − 1.00·17-s − 7.32·18-s + 0.847·19-s + 18.7·20-s − 2.78·21-s + 2.88·22-s + 2.43·23-s + 24.2·24-s − 1.24·25-s + 5.70·26-s − 2.78·27-s − 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 7529536 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7529536 ^{s/2} \, \Gamma_{\C}(s+49/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(7529536\) = \(2^{6} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.31031\times 10^{13}\) |
| Root analytic conductor: | \(14.5908\) |
| Motivic weight: | \(49\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 7529536,\ (\ :[49/2]^{6}),\ 1)\) |
Particular Values
| \(L(25)\) | \(\approx\) | \(1514.600379\) |
| \(L(\frac12)\) | \(\approx\) | \(1514.600379\) |
| \(L(\frac{51}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{24} T )^{6} \) |
| 7 | \( ( 1 + p^{24} T )^{6} \) | |
| good | 3 | \( 1 - 600288540488 T + \)\(28\!\cdots\!46\)\( p^{3} T^{2} - \)\(17\!\cdots\!96\)\( p^{7} T^{3} + \)\(19\!\cdots\!73\)\( p^{13} T^{4} - \)\(14\!\cdots\!36\)\( p^{21} T^{5} + \)\(14\!\cdots\!96\)\( p^{31} T^{6} - \)\(14\!\cdots\!36\)\( p^{70} T^{7} + \)\(19\!\cdots\!73\)\( p^{111} T^{8} - \)\(17\!\cdots\!96\)\( p^{154} T^{9} + \)\(28\!\cdots\!46\)\( p^{199} T^{10} - 600288540488 p^{245} T^{11} + p^{294} T^{12} \) |
| 5 | \( 1 - 237495319633852644 T + \)\(12\!\cdots\!54\)\( p^{4} T^{2} - \)\(89\!\cdots\!68\)\( p^{6} T^{3} + \)\(30\!\cdots\!07\)\( p^{10} T^{4} - \)\(27\!\cdots\!36\)\( p^{16} T^{5} + \)\(69\!\cdots\!16\)\( p^{20} T^{6} - \)\(27\!\cdots\!36\)\( p^{65} T^{7} + \)\(30\!\cdots\!07\)\( p^{108} T^{8} - \)\(89\!\cdots\!68\)\( p^{153} T^{9} + \)\(12\!\cdots\!54\)\( p^{200} T^{10} - 237495319633852644 p^{245} T^{11} + p^{294} T^{12} \) | |
| 11 | \( 1 - \)\(22\!\cdots\!08\)\( T + \)\(12\!\cdots\!22\)\( p T^{2} + \)\(18\!\cdots\!40\)\( p^{3} T^{3} + \)\(33\!\cdots\!09\)\( p^{5} T^{4} - \)\(92\!\cdots\!40\)\( p^{9} T^{5} + \)\(63\!\cdots\!96\)\( p^{13} T^{6} - \)\(92\!\cdots\!40\)\( p^{58} T^{7} + \)\(33\!\cdots\!09\)\( p^{103} T^{8} + \)\(18\!\cdots\!40\)\( p^{150} T^{9} + \)\(12\!\cdots\!22\)\( p^{197} T^{10} - \)\(22\!\cdots\!08\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 13 | \( 1 - \)\(26\!\cdots\!24\)\( T + \)\(10\!\cdots\!78\)\( T^{2} - \)\(29\!\cdots\!80\)\( p T^{3} + \)\(31\!\cdots\!55\)\( p^{4} T^{4} - \)\(54\!\cdots\!68\)\( p^{5} T^{5} + \)\(59\!\cdots\!16\)\( p^{8} T^{6} - \)\(54\!\cdots\!68\)\( p^{54} T^{7} + \)\(31\!\cdots\!55\)\( p^{102} T^{8} - \)\(29\!\cdots\!80\)\( p^{148} T^{9} + \)\(10\!\cdots\!78\)\( p^{196} T^{10} - \)\(26\!\cdots\!24\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 17 | \( 1 + \)\(82\!\cdots\!00\)\( p T + \)\(10\!\cdots\!66\)\( T^{2} + \)\(72\!\cdots\!08\)\( p T^{3} + \)\(92\!\cdots\!07\)\( p^{3} T^{4} + \)\(31\!\cdots\!84\)\( p^{5} T^{5} + \)\(28\!\cdots\!32\)\( p^{7} T^{6} + \)\(31\!\cdots\!84\)\( p^{54} T^{7} + \)\(92\!\cdots\!07\)\( p^{101} T^{8} + \)\(72\!\cdots\!08\)\( p^{148} T^{9} + \)\(10\!\cdots\!66\)\( p^{196} T^{10} + \)\(82\!\cdots\!00\)\( p^{246} T^{11} + p^{294} T^{12} \) | |
| 19 | \( 1 - \)\(18\!\cdots\!28\)\( T + \)\(80\!\cdots\!98\)\( T^{2} - \)\(39\!\cdots\!48\)\( p T^{3} + \)\(10\!\cdots\!35\)\( p^{2} T^{4} + \)\(18\!\cdots\!36\)\( p^{4} T^{5} + \)\(42\!\cdots\!12\)\( p^{6} T^{6} + \)\(18\!\cdots\!36\)\( p^{53} T^{7} + \)\(10\!\cdots\!35\)\( p^{100} T^{8} - \)\(39\!\cdots\!48\)\( p^{148} T^{9} + \)\(80\!\cdots\!98\)\( p^{196} T^{10} - \)\(18\!\cdots\!28\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 23 | \( 1 - \)\(56\!\cdots\!00\)\( T + \)\(16\!\cdots\!86\)\( p T^{2} - \)\(26\!\cdots\!00\)\( p^{2} T^{3} + \)\(43\!\cdots\!05\)\( p^{3} T^{4} - \)\(21\!\cdots\!00\)\( p^{5} T^{5} + \)\(11\!\cdots\!20\)\( p^{7} T^{6} - \)\(21\!\cdots\!00\)\( p^{54} T^{7} + \)\(43\!\cdots\!05\)\( p^{101} T^{8} - \)\(26\!\cdots\!00\)\( p^{149} T^{9} + \)\(16\!\cdots\!86\)\( p^{197} T^{10} - \)\(56\!\cdots\!00\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 29 | \( 1 + \)\(48\!\cdots\!32\)\( T + \)\(96\!\cdots\!98\)\( T^{2} + \)\(16\!\cdots\!84\)\( p T^{3} + \)\(76\!\cdots\!43\)\( p^{2} T^{4} + \)\(10\!\cdots\!76\)\( p^{3} T^{5} + \)\(50\!\cdots\!72\)\( p^{4} T^{6} + \)\(10\!\cdots\!76\)\( p^{52} T^{7} + \)\(76\!\cdots\!43\)\( p^{100} T^{8} + \)\(16\!\cdots\!84\)\( p^{148} T^{9} + \)\(96\!\cdots\!98\)\( p^{196} T^{10} + \)\(48\!\cdots\!32\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 31 | \( 1 - \)\(64\!\cdots\!80\)\( T + \)\(54\!\cdots\!50\)\( T^{2} - \)\(89\!\cdots\!76\)\( p T^{3} + \)\(15\!\cdots\!15\)\( p^{2} T^{4} - \)\(19\!\cdots\!80\)\( p^{3} T^{5} + \)\(24\!\cdots\!36\)\( p^{4} T^{6} - \)\(19\!\cdots\!80\)\( p^{52} T^{7} + \)\(15\!\cdots\!15\)\( p^{100} T^{8} - \)\(89\!\cdots\!76\)\( p^{148} T^{9} + \)\(54\!\cdots\!50\)\( p^{196} T^{10} - \)\(64\!\cdots\!80\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 37 | \( 1 + \)\(17\!\cdots\!68\)\( T + \)\(51\!\cdots\!98\)\( p T^{2} + \)\(31\!\cdots\!48\)\( p^{2} T^{3} + \)\(42\!\cdots\!51\)\( p^{3} T^{4} + \)\(29\!\cdots\!32\)\( p^{4} T^{5} + \)\(24\!\cdots\!60\)\( p^{5} T^{6} + \)\(29\!\cdots\!32\)\( p^{53} T^{7} + \)\(42\!\cdots\!51\)\( p^{101} T^{8} + \)\(31\!\cdots\!48\)\( p^{149} T^{9} + \)\(51\!\cdots\!98\)\( p^{197} T^{10} + \)\(17\!\cdots\!68\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 41 | \( 1 - \)\(20\!\cdots\!88\)\( p T + \)\(21\!\cdots\!22\)\( p^{2} T^{2} - \)\(21\!\cdots\!04\)\( p^{3} T^{3} + \)\(24\!\cdots\!95\)\( p^{4} T^{4} - \)\(11\!\cdots\!32\)\( p^{5} T^{5} + \)\(18\!\cdots\!12\)\( p^{6} T^{6} - \)\(11\!\cdots\!32\)\( p^{54} T^{7} + \)\(24\!\cdots\!95\)\( p^{102} T^{8} - \)\(21\!\cdots\!04\)\( p^{150} T^{9} + \)\(21\!\cdots\!22\)\( p^{198} T^{10} - \)\(20\!\cdots\!88\)\( p^{246} T^{11} + p^{294} T^{12} \) | |
| 43 | \( 1 - \)\(40\!\cdots\!32\)\( p T + \)\(27\!\cdots\!98\)\( p^{2} T^{2} - \)\(73\!\cdots\!36\)\( p^{3} T^{3} + \)\(31\!\cdots\!59\)\( p^{4} T^{4} - \)\(64\!\cdots\!16\)\( p^{5} T^{5} + \)\(21\!\cdots\!72\)\( p^{6} T^{6} - \)\(64\!\cdots\!16\)\( p^{54} T^{7} + \)\(31\!\cdots\!59\)\( p^{102} T^{8} - \)\(73\!\cdots\!36\)\( p^{150} T^{9} + \)\(27\!\cdots\!98\)\( p^{198} T^{10} - \)\(40\!\cdots\!32\)\( p^{246} T^{11} + p^{294} T^{12} \) | |
| 47 | \( 1 - \)\(39\!\cdots\!52\)\( p T + \)\(90\!\cdots\!62\)\( p^{2} T^{2} - \)\(88\!\cdots\!96\)\( p^{3} T^{3} - \)\(10\!\cdots\!73\)\( p^{4} T^{4} + \)\(66\!\cdots\!64\)\( p^{5} T^{5} - \)\(14\!\cdots\!44\)\( p^{6} T^{6} + \)\(66\!\cdots\!64\)\( p^{54} T^{7} - \)\(10\!\cdots\!73\)\( p^{102} T^{8} - \)\(88\!\cdots\!96\)\( p^{150} T^{9} + \)\(90\!\cdots\!62\)\( p^{198} T^{10} - \)\(39\!\cdots\!52\)\( p^{246} T^{11} + p^{294} T^{12} \) | |
| 53 | \( 1 - \)\(23\!\cdots\!72\)\( T + \)\(15\!\cdots\!74\)\( T^{2} - \)\(28\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!23\)\( T^{4} - \)\(15\!\cdots\!44\)\( T^{5} + \)\(43\!\cdots\!24\)\( T^{6} - \)\(15\!\cdots\!44\)\( p^{49} T^{7} + \)\(10\!\cdots\!23\)\( p^{98} T^{8} - \)\(28\!\cdots\!56\)\( p^{147} T^{9} + \)\(15\!\cdots\!74\)\( p^{196} T^{10} - \)\(23\!\cdots\!72\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 59 | \( 1 - \)\(30\!\cdots\!20\)\( T + \)\(14\!\cdots\!38\)\( T^{2} - \)\(58\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!83\)\( T^{4} - \)\(45\!\cdots\!36\)\( T^{5} + \)\(13\!\cdots\!60\)\( T^{6} - \)\(45\!\cdots\!36\)\( p^{49} T^{7} + \)\(16\!\cdots\!83\)\( p^{98} T^{8} - \)\(58\!\cdots\!64\)\( p^{147} T^{9} + \)\(14\!\cdots\!38\)\( p^{196} T^{10} - \)\(30\!\cdots\!20\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 61 | \( 1 - \)\(15\!\cdots\!44\)\( T + \)\(25\!\cdots\!10\)\( T^{2} - \)\(23\!\cdots\!28\)\( T^{3} + \)\(21\!\cdots\!67\)\( T^{4} - \)\(22\!\cdots\!04\)\( p T^{5} + \)\(90\!\cdots\!92\)\( T^{6} - \)\(22\!\cdots\!04\)\( p^{50} T^{7} + \)\(21\!\cdots\!67\)\( p^{98} T^{8} - \)\(23\!\cdots\!28\)\( p^{147} T^{9} + \)\(25\!\cdots\!10\)\( p^{196} T^{10} - \)\(15\!\cdots\!44\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 67 | \( 1 + \)\(12\!\cdots\!28\)\( T + \)\(94\!\cdots\!58\)\( T^{2} - \)\(11\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!79\)\( T^{4} - \)\(14\!\cdots\!36\)\( T^{5} + \)\(10\!\cdots\!32\)\( T^{6} - \)\(14\!\cdots\!36\)\( p^{49} T^{7} + \)\(37\!\cdots\!79\)\( p^{98} T^{8} - \)\(11\!\cdots\!96\)\( p^{147} T^{9} + \)\(94\!\cdots\!58\)\( p^{196} T^{10} + \)\(12\!\cdots\!28\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 71 | \( 1 - \)\(28\!\cdots\!08\)\( T + \)\(14\!\cdots\!62\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!39\)\( T^{4} + \)\(59\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!36\)\( T^{6} + \)\(59\!\cdots\!00\)\( p^{49} T^{7} + \)\(75\!\cdots\!39\)\( p^{98} T^{8} + \)\(14\!\cdots\!20\)\( p^{147} T^{9} + \)\(14\!\cdots\!62\)\( p^{196} T^{10} - \)\(28\!\cdots\!08\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 73 | \( 1 - \)\(63\!\cdots\!76\)\( T + \)\(78\!\cdots\!94\)\( T^{2} - \)\(35\!\cdots\!12\)\( T^{3} + \)\(30\!\cdots\!31\)\( T^{4} - \)\(11\!\cdots\!44\)\( T^{5} + \)\(76\!\cdots\!52\)\( T^{6} - \)\(11\!\cdots\!44\)\( p^{49} T^{7} + \)\(30\!\cdots\!31\)\( p^{98} T^{8} - \)\(35\!\cdots\!12\)\( p^{147} T^{9} + \)\(78\!\cdots\!94\)\( p^{196} T^{10} - \)\(63\!\cdots\!76\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 79 | \( 1 + \)\(67\!\cdots\!64\)\( T + \)\(23\!\cdots\!30\)\( T^{2} + \)\(46\!\cdots\!68\)\( T^{3} + \)\(14\!\cdots\!55\)\( T^{4} + \)\(94\!\cdots\!80\)\( T^{5} + \)\(39\!\cdots\!32\)\( T^{6} + \)\(94\!\cdots\!80\)\( p^{49} T^{7} + \)\(14\!\cdots\!55\)\( p^{98} T^{8} + \)\(46\!\cdots\!68\)\( p^{147} T^{9} + \)\(23\!\cdots\!30\)\( p^{196} T^{10} + \)\(67\!\cdots\!64\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 83 | \( 1 + \)\(45\!\cdots\!60\)\( T + \)\(37\!\cdots\!02\)\( T^{2} + \)\(39\!\cdots\!24\)\( T^{3} + \)\(60\!\cdots\!27\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{5} + \)\(68\!\cdots\!84\)\( T^{6} - \)\(11\!\cdots\!16\)\( p^{49} T^{7} + \)\(60\!\cdots\!27\)\( p^{98} T^{8} + \)\(39\!\cdots\!24\)\( p^{147} T^{9} + \)\(37\!\cdots\!02\)\( p^{196} T^{10} + \)\(45\!\cdots\!60\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 89 | \( 1 - \)\(80\!\cdots\!96\)\( T + \)\(21\!\cdots\!50\)\( T^{2} - \)\(13\!\cdots\!96\)\( T^{3} + \)\(18\!\cdots\!67\)\( T^{4} - \)\(86\!\cdots\!72\)\( T^{5} + \)\(81\!\cdots\!64\)\( T^{6} - \)\(86\!\cdots\!72\)\( p^{49} T^{7} + \)\(18\!\cdots\!67\)\( p^{98} T^{8} - \)\(13\!\cdots\!96\)\( p^{147} T^{9} + \)\(21\!\cdots\!50\)\( p^{196} T^{10} - \)\(80\!\cdots\!96\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
| 97 | \( 1 - \)\(22\!\cdots\!24\)\( T + \)\(58\!\cdots\!86\)\( T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!47\)\( T^{4} - \)\(49\!\cdots\!40\)\( T^{5} + \)\(60\!\cdots\!16\)\( T^{6} - \)\(49\!\cdots\!40\)\( p^{49} T^{7} + \)\(22\!\cdots\!47\)\( p^{98} T^{8} - \)\(15\!\cdots\!20\)\( p^{147} T^{9} + \)\(58\!\cdots\!86\)\( p^{196} T^{10} - \)\(22\!\cdots\!24\)\( p^{245} T^{11} + p^{294} T^{12} \) | |
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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
−5.04775263835487492349885414580, −4.27044333467409475948014725103, −4.19879830932058351566376318980, −4.06103302684063759850543689968, −3.96148255252160024479821106679, −3.90085191813886207117298311163, −3.78278866081844161587095358385, −3.23518046144624203389066997807, −3.15781888198153297835241578775, −3.13522363421283732653278690868, −2.88019931043785116701525379255, −2.81826429340818369728564089893, −2.81092749192979797400289676292, −2.28236763988133352701691328181, −2.23998003477643538591394338025, −2.15044158838848237406064203462, −2.11574865973856595285137194992, −1.77786963773334805215347719723, −1.54294341681577836996148318175, −1.15689489829813676536423483943, −1.05667108920625405762990222950, −0.824547200414747841575017382296, −0.56580599530273756148940753427, −0.51503788565814098790388542119, −0.35243240421970505006579977209, 0.35243240421970505006579977209, 0.51503788565814098790388542119, 0.56580599530273756148940753427, 0.824547200414747841575017382296, 1.05667108920625405762990222950, 1.15689489829813676536423483943, 1.54294341681577836996148318175, 1.77786963773334805215347719723, 2.11574865973856595285137194992, 2.15044158838848237406064203462, 2.23998003477643538591394338025, 2.28236763988133352701691328181, 2.81092749192979797400289676292, 2.81826429340818369728564089893, 2.88019931043785116701525379255, 3.13522363421283732653278690868, 3.15781888198153297835241578775, 3.23518046144624203389066997807, 3.78278866081844161587095358385, 3.90085191813886207117298311163, 3.96148255252160024479821106679, 4.06103302684063759850543689968, 4.19879830932058351566376318980, 4.27044333467409475948014725103, 5.04775263835487492349885414580
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.