Dirichlet series
| L(s) = 1 | − 6.16e5·3-s + 1.86e7·4-s + 1.98e9·7-s − 2.05e11·9-s − 1.15e13·12-s − 7.35e13·13-s − 1.99e14·16-s + 6.09e15·19-s − 1.22e15·21-s + 7.56e16·25-s + 1.53e17·27-s + 3.70e16·28-s − 1.13e18·31-s − 3.83e18·36-s − 1.40e19·37-s + 4.53e19·39-s − 1.14e20·43-s + 1.23e20·48-s − 5.18e20·49-s − 1.37e21·52-s − 3.75e21·57-s + 1.00e22·61-s − 4.08e20·63-s − 5.15e21·64-s − 5.44e21·67-s + 7.22e21·73-s − 4.66e22·75-s + ⋯ |
| L(s) = 1 | − 1.16·3-s + 1.11·4-s + 0.143·7-s − 0.727·9-s − 1.29·12-s − 3.15·13-s − 0.710·16-s + 2.75·19-s − 0.166·21-s + 1.26·25-s + 1.02·27-s + 0.159·28-s − 1.44·31-s − 0.808·36-s − 2.13·37-s + 3.66·39-s − 2.86·43-s + 0.824·48-s − 2.70·49-s − 3.51·52-s − 3.19·57-s + 3.77·61-s − 0.104·63-s − 1.09·64-s − 0.665·67-s + 0.315·73-s − 1.47·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+12)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(729\) = \(3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.72286\times 10^{6}\) |
| Root analytic conductor: | \(3.30892\) |
| Motivic weight: | \(24\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 729,\ (\ :[12]^{6}),\ 1)\) |
Particular Values
| \(L(\frac{25}{2})\) | \(\approx\) | \(0.2274107466\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2274107466\) |
| \(L(13)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 22846 p^{3} T + 29763781 p^{9} T^{2} + 2589165508 p^{17} T^{3} + 29763781 p^{33} T^{4} + 22846 p^{51} T^{5} + p^{72} T^{6} \) |
| good | 2 | \( 1 - 2332281 p^{3} T^{2} + 535248848211 p^{10} T^{4} - 1049829664886923 p^{23} T^{6} + 535248848211 p^{58} T^{8} - 2332281 p^{99} T^{10} + p^{144} T^{12} \) |
| 5 | \( 1 - 605412656357502 p^{3} T^{2} + \)\(40\!\cdots\!63\)\( p^{6} T^{4} - \)\(29\!\cdots\!84\)\( p^{9} T^{6} + \)\(40\!\cdots\!63\)\( p^{54} T^{8} - 605412656357502 p^{99} T^{10} + p^{144} T^{12} \) | |
| 7 | \( ( 1 - 142004634 p T + 760580034374478921 p^{3} T^{2} - \)\(19\!\cdots\!24\)\( p^{6} T^{3} + 760580034374478921 p^{27} T^{4} - 142004634 p^{49} T^{5} + p^{72} T^{6} )^{2} \) | |
| 11 | \( 1 - \)\(25\!\cdots\!46\)\( p^{2} T^{2} + \)\(37\!\cdots\!95\)\( p^{4} T^{4} - \)\(30\!\cdots\!40\)\( p^{8} T^{6} + \)\(37\!\cdots\!95\)\( p^{52} T^{8} - \)\(25\!\cdots\!46\)\( p^{98} T^{10} + p^{144} T^{12} \) | |
| 13 | \( ( 1 + 36771219031962 T + \)\(11\!\cdots\!11\)\( p T^{2} + \)\(19\!\cdots\!56\)\( p^{2} T^{3} + \)\(11\!\cdots\!11\)\( p^{25} T^{4} + 36771219031962 p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
| 17 | \( 1 - \)\(12\!\cdots\!78\)\( T^{2} + \)\(28\!\cdots\!51\)\( p^{2} T^{4} - \)\(41\!\cdots\!64\)\( p^{4} T^{6} + \)\(28\!\cdots\!51\)\( p^{50} T^{8} - \)\(12\!\cdots\!78\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 19 | \( ( 1 - 160400512986162 p T + \)\(35\!\cdots\!99\)\( p^{2} T^{2} - \)\(39\!\cdots\!56\)\( p^{3} T^{3} + \)\(35\!\cdots\!99\)\( p^{26} T^{4} - 160400512986162 p^{49} T^{5} + p^{72} T^{6} )^{2} \) | |
| 23 | \( 1 - \)\(42\!\cdots\!82\)\( p^{2} T^{2} + \)\(84\!\cdots\!79\)\( p^{4} T^{4} - \)\(97\!\cdots\!16\)\( p^{6} T^{6} + \)\(84\!\cdots\!79\)\( p^{52} T^{8} - \)\(42\!\cdots\!82\)\( p^{98} T^{10} + p^{144} T^{12} \) | |
| 29 | \( 1 - \)\(51\!\cdots\!66\)\( T^{2} + \)\(12\!\cdots\!35\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!35\)\( p^{48} T^{8} - \)\(51\!\cdots\!66\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 31 | \( ( 1 + 568921955487610314 T + \)\(15\!\cdots\!95\)\( T^{2} + \)\(23\!\cdots\!60\)\( p T^{3} + \)\(15\!\cdots\!95\)\( p^{24} T^{4} + 568921955487610314 p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
| 37 | \( ( 1 + 7016311747859411802 T + \)\(12\!\cdots\!23\)\( T^{2} - \)\(25\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!23\)\( p^{24} T^{4} + 7016311747859411802 p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
| 41 | \( 1 - \)\(24\!\cdots\!86\)\( T^{2} + \)\(27\!\cdots\!95\)\( T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + \)\(27\!\cdots\!95\)\( p^{48} T^{8} - \)\(24\!\cdots\!86\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 43 | \( ( 1 + 1332596490099329694 p T + \)\(46\!\cdots\!03\)\( T^{2} + \)\(15\!\cdots\!84\)\( T^{3} + \)\(46\!\cdots\!03\)\( p^{24} T^{4} + 1332596490099329694 p^{49} T^{5} + p^{72} T^{6} )^{2} \) | |
| 47 | \( 1 - \)\(50\!\cdots\!78\)\( T^{2} + \)\(55\!\cdots\!91\)\( p^{2} T^{4} - \)\(40\!\cdots\!04\)\( p^{4} T^{6} + \)\(55\!\cdots\!91\)\( p^{50} T^{8} - \)\(50\!\cdots\!78\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 53 | \( 1 - \)\(11\!\cdots\!78\)\( T^{2} + \)\(59\!\cdots\!19\)\( T^{4} - \)\(18\!\cdots\!24\)\( T^{6} + \)\(59\!\cdots\!19\)\( p^{48} T^{8} - \)\(11\!\cdots\!78\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 59 | \( 1 - \)\(25\!\cdots\!86\)\( T^{2} + \)\(16\!\cdots\!95\)\( T^{4} - \)\(42\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!95\)\( p^{48} T^{8} - \)\(25\!\cdots\!86\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 61 | \( ( 1 - \)\(50\!\cdots\!26\)\( T + \)\(26\!\cdots\!15\)\( T^{2} - \)\(72\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!15\)\( p^{24} T^{4} - \)\(50\!\cdots\!26\)\( p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
| 67 | \( ( 1 + \)\(27\!\cdots\!42\)\( T + \)\(19\!\cdots\!43\)\( T^{2} + \)\(36\!\cdots\!64\)\( T^{3} + \)\(19\!\cdots\!43\)\( p^{24} T^{4} + \)\(27\!\cdots\!42\)\( p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
| 71 | \( 1 - \)\(94\!\cdots\!66\)\( T^{2} + \)\(42\!\cdots\!35\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!35\)\( p^{48} T^{8} - \)\(94\!\cdots\!66\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 73 | \( ( 1 - \)\(36\!\cdots\!98\)\( T + \)\(13\!\cdots\!63\)\( T^{2} - \)\(23\!\cdots\!36\)\( T^{3} + \)\(13\!\cdots\!63\)\( p^{24} T^{4} - \)\(36\!\cdots\!98\)\( p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
| 79 | \( ( 1 + \)\(21\!\cdots\!62\)\( T + \)\(23\!\cdots\!99\)\( T^{2} + \)\(17\!\cdots\!16\)\( T^{3} + \)\(23\!\cdots\!99\)\( p^{24} T^{4} + \)\(21\!\cdots\!62\)\( p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
| 83 | \( 1 - \)\(39\!\cdots\!98\)\( T^{2} + \)\(81\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} + \)\(81\!\cdots\!39\)\( p^{48} T^{8} - \)\(39\!\cdots\!98\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 89 | \( 1 - \)\(19\!\cdots\!66\)\( T^{2} + \)\(21\!\cdots\!95\)\( T^{4} - \)\(14\!\cdots\!40\)\( T^{6} + \)\(21\!\cdots\!95\)\( p^{48} T^{8} - \)\(19\!\cdots\!66\)\( p^{96} T^{10} + p^{144} T^{12} \) | |
| 97 | \( ( 1 + \)\(70\!\cdots\!22\)\( T + \)\(54\!\cdots\!63\)\( T^{2} + \)\(62\!\cdots\!64\)\( T^{3} + \)\(54\!\cdots\!63\)\( p^{24} T^{4} + \)\(70\!\cdots\!22\)\( p^{48} T^{5} + p^{72} T^{6} )^{2} \) | |
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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
−9.890131449050827775582157960062, −9.712623886490538871507430587072, −9.431985804039432003731389419273, −8.787316492280478381790775979968, −8.261846957802239371978399847218, −8.215421788442511990278485815282, −7.11665161871161613171616554627, −7.09964334032194855851020738887, −7.07245650441216570917958217340, −6.94938269562008727452259251303, −5.87481593353681542930751611414, −5.74211752795144848573644770440, −5.30562653959132689833891075882, −4.94145017193832196632629452459, −4.92472565597131897720461083388, −4.32693866840688886923668904176, −3.21506201222435282932798577571, −3.17224056139292118636646409737, −3.02855356145617913127565593212, −2.27006090983739967173445177654, −1.93949047850805886881082440041, −1.70153572843531184201168919727, −1.05468511780177358420819451744, −0.39189162539001633464885667364, −0.12072612718175976728327597425, 0.12072612718175976728327597425, 0.39189162539001633464885667364, 1.05468511780177358420819451744, 1.70153572843531184201168919727, 1.93949047850805886881082440041, 2.27006090983739967173445177654, 3.02855356145617913127565593212, 3.17224056139292118636646409737, 3.21506201222435282932798577571, 4.32693866840688886923668904176, 4.92472565597131897720461083388, 4.94145017193832196632629452459, 5.30562653959132689833891075882, 5.74211752795144848573644770440, 5.87481593353681542930751611414, 6.94938269562008727452259251303, 7.07245650441216570917958217340, 7.09964334032194855851020738887, 7.11665161871161613171616554627, 8.215421788442511990278485815282, 8.261846957802239371978399847218, 8.787316492280478381790775979968, 9.431985804039432003731389419273, 9.712623886490538871507430587072, 9.890131449050827775582157960062
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.