Dirichlet series
| L(s) = 1 | + 1.37e10·2-s − 3.33e16·3-s − 1.17e20·4-s − 1.80e23·5-s − 4.58e26·6-s − 2.80e28·7-s − 3.09e30·8-s + 6.48e32·9-s − 2.47e33·10-s + 1.25e33·11-s + 3.92e36·12-s − 1.03e36·13-s − 3.85e38·14-s + 6.01e39·15-s − 2.42e40·16-s − 2.23e40·17-s + 8.91e42·18-s − 6.70e42·19-s + 2.12e43·20-s + 9.36e44·21-s + 1.72e43·22-s + 1.22e46·23-s + 1.03e47·24-s − 1.24e47·25-s − 1.41e46·26-s − 9.62e48·27-s + 3.30e48·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 3.46·3-s − 0.798·4-s − 0.692·5-s − 3.91·6-s − 1.37·7-s − 1.72·8-s + 7·9-s − 0.783·10-s + 0.0162·11-s + 2.76·12-s − 0.0497·13-s − 1.55·14-s + 2.39·15-s − 1.11·16-s − 0.134·17-s + 7.91·18-s − 0.973·19-s + 0.552·20-s + 4.75·21-s + 0.0184·22-s + 2.95·23-s + 5.98·24-s − 1.83·25-s − 0.0562·26-s − 10.7·27-s + 1.09·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+67/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(729\) = \(3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.84857\times 10^{11}\) |
| Root analytic conductor: | \(9.23510\) |
| Motivic weight: | \(67\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 729,\ (\ :[67/2]^{6}),\ 1)\) |
Particular Values
| \(L(34)\) | \(\approx\) | \(6.051912692\times10^{-5}\) |
| \(L(\frac12)\) | \(\approx\) | \(6.051912692\times10^{-5}\) |
| \(L(\frac{69}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{33} T )^{6} \) |
| 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
−5.72967124991006947899200807819, −5.31092524522969633570688180997, −5.26734437045866345465748285342, −5.06336051497060170207912828714, −5.05585809684835227487684422942, −4.68190480548827426535276634282, −4.48418303688646061563989081238, −4.02045255223821495990324202174, −3.94894473150087184723454794389, −3.89642223838088014899457772704, −3.69685319425590307447665814527, −3.56675364695678410427858841213, −3.07186828792486726105567966000, −2.70633131761827491738273166491, −2.59996056363700573207562719676, −2.21130381419611809885876262625, −1.85811698967749958776715425937, −1.65165640875748098088306502292, −1.47339164585099778097788536703, −1.10761350216465096631539876515, −0.829137807603112330932751473845, −0.62049057381001765270181277787, −0.45732055740112826516995209584, −0.18059237319261700147618879565, −0.00296699469778585821493152502, 0.00296699469778585821493152502, 0.18059237319261700147618879565, 0.45732055740112826516995209584, 0.62049057381001765270181277787, 0.829137807603112330932751473845, 1.10761350216465096631539876515, 1.47339164585099778097788536703, 1.65165640875748098088306502292, 1.85811698967749958776715425937, 2.21130381419611809885876262625, 2.59996056363700573207562719676, 2.70633131761827491738273166491, 3.07186828792486726105567966000, 3.56675364695678410427858841213, 3.69685319425590307447665814527, 3.89642223838088014899457772704, 3.94894473150087184723454794389, 4.02045255223821495990324202174, 4.48418303688646061563989081238, 4.68190480548827426535276634282, 5.05585809684835227487684422942, 5.06336051497060170207912828714, 5.26734437045866345465748285342, 5.31092524522969633570688180997, 5.72967124991006947899200807819
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.