Dirichlet series
| L(s) = 1 | + 5.70e10·2-s − 2.54e22·4-s + 3.89e25·5-s + 1.92e30·7-s − 3.70e33·8-s + 2.22e36·10-s + 9.45e38·11-s + 5.33e41·13-s + 1.09e41·14-s − 7.63e44·16-s − 1.82e46·17-s + 1.06e48·19-s − 9.90e47·20-s + 5.39e49·22-s − 1.51e51·23-s − 6.89e52·25-s + 3.04e52·26-s − 4.88e52·28-s − 1.47e55·29-s − 4.17e55·31-s − 1.80e56·32-s − 1.04e57·34-s + 7.50e55·35-s + 9.85e58·37-s + 6.05e58·38-s − 1.44e59·40-s − 5.03e60·41-s + ⋯ |
| L(s) = 1 | + 0.293·2-s − 0.672·4-s + 0.239·5-s + 0.0391·7-s − 0.505·8-s + 0.0703·10-s + 0.838·11-s + 0.899·13-s + 0.0115·14-s − 0.534·16-s − 1.31·17-s + 1.18·19-s − 0.161·20-s + 0.246·22-s − 1.30·23-s − 2.60·25-s + 0.264·26-s − 0.0263·28-s − 2.12·29-s − 0.494·31-s − 0.650·32-s − 0.386·34-s + 0.00938·35-s + 1.53·37-s + 0.347·38-s − 0.121·40-s − 1.67·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+75/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(531441\) = \(3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.08599\times 10^{15}\) |
| Root analytic conductor: | \(17.9054\) |
| Motivic weight: | \(75\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 531441,\ (\ :[75/2]^{6}),\ 1)\) |
Particular Values
| \(L(38)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{77}{2})\) | 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 | \( 1 - 7135102755 p^{3} T + 13996910454673606305 p^{11} T^{2} + \)\(14\!\cdots\!45\)\( p^{22} T^{3} + \)\(45\!\cdots\!13\)\( p^{38} T^{4} + \)\(65\!\cdots\!45\)\( p^{58} T^{5} + \)\(23\!\cdots\!45\)\( p^{80} T^{6} + \)\(65\!\cdots\!45\)\( p^{133} T^{7} + \)\(45\!\cdots\!13\)\( p^{188} T^{8} + \)\(14\!\cdots\!45\)\( p^{247} T^{9} + 13996910454673606305 p^{311} T^{10} - 7135102755 p^{378} T^{11} + p^{450} T^{12} \) |
| 5 | \( 1 - \)\(77\!\cdots\!88\)\( p T + \)\(22\!\cdots\!94\)\( p^{5} T^{2} - \)\(11\!\cdots\!64\)\( p^{10} T^{3} + \)\(37\!\cdots\!47\)\( p^{17} T^{4} - \)\(41\!\cdots\!48\)\( p^{26} T^{5} + \)\(61\!\cdots\!44\)\( p^{36} T^{6} - \)\(41\!\cdots\!48\)\( p^{101} T^{7} + \)\(37\!\cdots\!47\)\( p^{167} T^{8} - \)\(11\!\cdots\!64\)\( p^{235} T^{9} + \)\(22\!\cdots\!94\)\( p^{305} T^{10} - \)\(77\!\cdots\!88\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 7 | \( 1 - \)\(27\!\cdots\!00\)\( p T + \)\(42\!\cdots\!50\)\( p^{4} T^{2} + \)\(16\!\cdots\!00\)\( p^{9} T^{3} + \)\(10\!\cdots\!29\)\( p^{15} T^{4} + \)\(11\!\cdots\!00\)\( p^{22} T^{5} + \)\(89\!\cdots\!00\)\( p^{31} T^{6} + \)\(11\!\cdots\!00\)\( p^{97} T^{7} + \)\(10\!\cdots\!29\)\( p^{165} T^{8} + \)\(16\!\cdots\!00\)\( p^{234} T^{9} + \)\(42\!\cdots\!50\)\( p^{304} T^{10} - \)\(27\!\cdots\!00\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 11 | \( 1 - \)\(85\!\cdots\!08\)\( p T + \)\(44\!\cdots\!86\)\( p^{3} T^{2} - \)\(29\!\cdots\!80\)\( p^{5} T^{3} + \)\(75\!\cdots\!95\)\( p^{8} T^{4} - \)\(31\!\cdots\!68\)\( p^{13} T^{5} + \)\(42\!\cdots\!64\)\( p^{19} T^{6} - \)\(31\!\cdots\!68\)\( p^{88} T^{7} + \)\(75\!\cdots\!95\)\( p^{158} T^{8} - \)\(29\!\cdots\!80\)\( p^{230} T^{9} + \)\(44\!\cdots\!86\)\( p^{303} T^{10} - \)\(85\!\cdots\!08\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 13 | \( 1 - \)\(41\!\cdots\!40\)\( p T + \)\(26\!\cdots\!70\)\( p^{5} T^{2} - \)\(19\!\cdots\!20\)\( p^{3} T^{3} + \)\(56\!\cdots\!83\)\( p^{6} T^{4} - \)\(10\!\cdots\!40\)\( p^{10} T^{5} + \)\(74\!\cdots\!40\)\( p^{15} T^{6} - \)\(10\!\cdots\!40\)\( p^{85} T^{7} + \)\(56\!\cdots\!83\)\( p^{156} T^{8} - \)\(19\!\cdots\!20\)\( p^{228} T^{9} + \)\(26\!\cdots\!70\)\( p^{305} T^{10} - \)\(41\!\cdots\!40\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 17 | \( 1 + \)\(18\!\cdots\!80\)\( T + \)\(59\!\cdots\!10\)\( p T^{2} + \)\(28\!\cdots\!80\)\( p^{3} T^{3} + \)\(17\!\cdots\!63\)\( p^{6} T^{4} + \)\(40\!\cdots\!20\)\( p^{9} T^{5} + \)\(10\!\cdots\!80\)\( p^{13} T^{6} + \)\(40\!\cdots\!20\)\( p^{84} T^{7} + \)\(17\!\cdots\!63\)\( p^{156} T^{8} + \)\(28\!\cdots\!80\)\( p^{228} T^{9} + \)\(59\!\cdots\!10\)\( p^{301} T^{10} + \)\(18\!\cdots\!80\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 19 | \( 1 - \)\(29\!\cdots\!80\)\( p^{2} T + \)\(67\!\cdots\!54\)\( p^{2} T^{2} - \)\(33\!\cdots\!00\)\( p^{3} T^{3} + \)\(27\!\cdots\!15\)\( p^{4} T^{4} - \)\(31\!\cdots\!00\)\( p^{7} T^{5} + \)\(56\!\cdots\!80\)\( p^{10} T^{6} - \)\(31\!\cdots\!00\)\( p^{82} T^{7} + \)\(27\!\cdots\!15\)\( p^{154} T^{8} - \)\(33\!\cdots\!00\)\( p^{228} T^{9} + \)\(67\!\cdots\!54\)\( p^{302} T^{10} - \)\(29\!\cdots\!80\)\( p^{377} T^{11} + p^{450} T^{12} \) | |
| 23 | \( 1 + \)\(65\!\cdots\!60\)\( p T + \)\(11\!\cdots\!10\)\( p^{2} T^{2} + \)\(22\!\cdots\!60\)\( p^{4} T^{3} + \)\(11\!\cdots\!23\)\( p^{6} T^{4} + \)\(76\!\cdots\!80\)\( p^{9} T^{5} + \)\(12\!\cdots\!20\)\( p^{12} T^{6} + \)\(76\!\cdots\!80\)\( p^{84} T^{7} + \)\(11\!\cdots\!23\)\( p^{156} T^{8} + \)\(22\!\cdots\!60\)\( p^{229} T^{9} + \)\(11\!\cdots\!10\)\( p^{302} T^{10} + \)\(65\!\cdots\!60\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 29 | \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(10\!\cdots\!86\)\( p T^{2} + \)\(35\!\cdots\!00\)\( p^{2} T^{3} + \)\(14\!\cdots\!35\)\( p^{3} T^{4} + \)\(12\!\cdots\!00\)\( p^{5} T^{5} + \)\(12\!\cdots\!20\)\( p^{7} T^{6} + \)\(12\!\cdots\!00\)\( p^{80} T^{7} + \)\(14\!\cdots\!35\)\( p^{153} T^{8} + \)\(35\!\cdots\!00\)\( p^{227} T^{9} + \)\(10\!\cdots\!86\)\( p^{301} T^{10} + \)\(14\!\cdots\!20\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 31 | \( 1 + \)\(41\!\cdots\!88\)\( T + \)\(51\!\cdots\!86\)\( p T^{2} + \)\(77\!\cdots\!80\)\( p^{2} T^{3} + \)\(17\!\cdots\!95\)\( p^{4} T^{4} + \)\(11\!\cdots\!68\)\( p^{6} T^{5} + \)\(15\!\cdots\!64\)\( p^{8} T^{6} + \)\(11\!\cdots\!68\)\( p^{81} T^{7} + \)\(17\!\cdots\!95\)\( p^{154} T^{8} + \)\(77\!\cdots\!80\)\( p^{227} T^{9} + \)\(51\!\cdots\!86\)\( p^{301} T^{10} + \)\(41\!\cdots\!88\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 37 | \( 1 - \)\(26\!\cdots\!20\)\( p T + \)\(11\!\cdots\!90\)\( p^{2} T^{2} - \)\(28\!\cdots\!40\)\( p^{3} T^{3} + \)\(74\!\cdots\!27\)\( p^{4} T^{4} - \)\(13\!\cdots\!60\)\( p^{5} T^{5} + \)\(29\!\cdots\!20\)\( p^{6} T^{6} - \)\(13\!\cdots\!60\)\( p^{80} T^{7} + \)\(74\!\cdots\!27\)\( p^{154} T^{8} - \)\(28\!\cdots\!40\)\( p^{228} T^{9} + \)\(11\!\cdots\!90\)\( p^{302} T^{10} - \)\(26\!\cdots\!20\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 41 | \( 1 + \)\(12\!\cdots\!32\)\( p T + \)\(29\!\cdots\!66\)\( T^{2} + \)\(66\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!95\)\( p T^{4} + \)\(11\!\cdots\!32\)\( p^{2} T^{5} + \)\(18\!\cdots\!44\)\( p^{3} T^{6} + \)\(11\!\cdots\!32\)\( p^{77} T^{7} + \)\(59\!\cdots\!95\)\( p^{151} T^{8} + \)\(66\!\cdots\!20\)\( p^{225} T^{9} + \)\(29\!\cdots\!66\)\( p^{300} T^{10} + \)\(12\!\cdots\!32\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 43 | \( 1 - \)\(27\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(40\!\cdots\!00\)\( p T^{3} + \)\(28\!\cdots\!03\)\( p^{2} T^{4} - \)\(10\!\cdots\!00\)\( p^{3} T^{5} + \)\(60\!\cdots\!00\)\( p^{4} T^{6} - \)\(10\!\cdots\!00\)\( p^{78} T^{7} + \)\(28\!\cdots\!03\)\( p^{152} T^{8} - \)\(40\!\cdots\!00\)\( p^{226} T^{9} + \)\(10\!\cdots\!50\)\( p^{300} T^{10} - \)\(27\!\cdots\!00\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 47 | \( 1 - \)\(13\!\cdots\!80\)\( T + \)\(16\!\cdots\!30\)\( T^{2} - \)\(30\!\cdots\!20\)\( p T^{3} + \)\(47\!\cdots\!83\)\( p^{2} T^{4} - \)\(62\!\cdots\!80\)\( p^{3} T^{5} + \)\(72\!\cdots\!40\)\( p^{4} T^{6} - \)\(62\!\cdots\!80\)\( p^{78} T^{7} + \)\(47\!\cdots\!83\)\( p^{152} T^{8} - \)\(30\!\cdots\!20\)\( p^{226} T^{9} + \)\(16\!\cdots\!30\)\( p^{300} T^{10} - \)\(13\!\cdots\!80\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 53 | \( 1 + \)\(64\!\cdots\!60\)\( T + \)\(76\!\cdots\!30\)\( T^{2} + \)\(54\!\cdots\!40\)\( p T^{3} + \)\(64\!\cdots\!83\)\( p^{2} T^{4} + \)\(25\!\cdots\!40\)\( p^{3} T^{5} + \)\(34\!\cdots\!40\)\( p^{4} T^{6} + \)\(25\!\cdots\!40\)\( p^{78} T^{7} + \)\(64\!\cdots\!83\)\( p^{152} T^{8} + \)\(54\!\cdots\!40\)\( p^{226} T^{9} + \)\(76\!\cdots\!30\)\( p^{300} T^{10} + \)\(64\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 59 | \( 1 - \)\(24\!\cdots\!60\)\( T + \)\(35\!\cdots\!66\)\( p T^{2} - \)\(17\!\cdots\!00\)\( p^{2} T^{3} + \)\(12\!\cdots\!85\)\( p^{3} T^{4} - \)\(49\!\cdots\!00\)\( p^{4} T^{5} + \)\(51\!\cdots\!80\)\( p^{6} T^{6} - \)\(49\!\cdots\!00\)\( p^{79} T^{7} + \)\(12\!\cdots\!85\)\( p^{153} T^{8} - \)\(17\!\cdots\!00\)\( p^{227} T^{9} + \)\(35\!\cdots\!66\)\( p^{301} T^{10} - \)\(24\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 61 | \( 1 + \)\(25\!\cdots\!88\)\( T + \)\(90\!\cdots\!06\)\( p T^{2} + \)\(22\!\cdots\!80\)\( p^{2} T^{3} + \)\(50\!\cdots\!95\)\( p^{3} T^{4} + \)\(90\!\cdots\!88\)\( p^{4} T^{5} + \)\(14\!\cdots\!24\)\( p^{5} T^{6} + \)\(90\!\cdots\!88\)\( p^{79} T^{7} + \)\(50\!\cdots\!95\)\( p^{153} T^{8} + \)\(22\!\cdots\!80\)\( p^{227} T^{9} + \)\(90\!\cdots\!06\)\( p^{301} T^{10} + \)\(25\!\cdots\!88\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 67 | \( 1 - \)\(14\!\cdots\!40\)\( p T + \)\(16\!\cdots\!30\)\( p^{2} T^{2} - \)\(12\!\cdots\!80\)\( p^{3} T^{3} + \)\(83\!\cdots\!07\)\( p^{4} T^{4} - \)\(44\!\cdots\!20\)\( p^{5} T^{5} + \)\(22\!\cdots\!40\)\( p^{6} T^{6} - \)\(44\!\cdots\!20\)\( p^{80} T^{7} + \)\(83\!\cdots\!07\)\( p^{154} T^{8} - \)\(12\!\cdots\!80\)\( p^{228} T^{9} + \)\(16\!\cdots\!30\)\( p^{302} T^{10} - \)\(14\!\cdots\!40\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 71 | \( 1 - \)\(36\!\cdots\!28\)\( p T + \)\(47\!\cdots\!26\)\( p^{2} T^{2} - \)\(44\!\cdots\!80\)\( p^{3} T^{3} + \)\(12\!\cdots\!95\)\( p^{4} T^{4} - \)\(10\!\cdots\!08\)\( p^{5} T^{5} + \)\(21\!\cdots\!44\)\( p^{6} T^{6} - \)\(10\!\cdots\!08\)\( p^{80} T^{7} + \)\(12\!\cdots\!95\)\( p^{154} T^{8} - \)\(44\!\cdots\!80\)\( p^{228} T^{9} + \)\(47\!\cdots\!26\)\( p^{302} T^{10} - \)\(36\!\cdots\!28\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 73 | \( 1 + \)\(41\!\cdots\!40\)\( p T + \)\(11\!\cdots\!10\)\( p^{2} T^{2} + \)\(20\!\cdots\!20\)\( p^{3} T^{3} + \)\(31\!\cdots\!67\)\( p^{4} T^{4} + \)\(40\!\cdots\!20\)\( p^{5} T^{5} + \)\(43\!\cdots\!80\)\( p^{6} T^{6} + \)\(40\!\cdots\!20\)\( p^{80} T^{7} + \)\(31\!\cdots\!67\)\( p^{154} T^{8} + \)\(20\!\cdots\!20\)\( p^{228} T^{9} + \)\(11\!\cdots\!10\)\( p^{302} T^{10} + \)\(41\!\cdots\!40\)\( p^{376} T^{11} + p^{450} T^{12} \) | |
| 79 | \( 1 - \)\(11\!\cdots\!20\)\( T + \)\(92\!\cdots\!94\)\( T^{2} - \)\(90\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(33\!\cdots\!00\)\( p^{75} T^{7} + \)\(40\!\cdots\!15\)\( p^{150} T^{8} - \)\(90\!\cdots\!00\)\( p^{225} T^{9} + \)\(92\!\cdots\!94\)\( p^{300} T^{10} - \)\(11\!\cdots\!20\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 83 | \( 1 - \)\(79\!\cdots\!60\)\( T + \)\(37\!\cdots\!70\)\( T^{2} - \)\(25\!\cdots\!20\)\( T^{3} + \)\(64\!\cdots\!47\)\( T^{4} - \)\(36\!\cdots\!80\)\( T^{5} + \)\(67\!\cdots\!60\)\( T^{6} - \)\(36\!\cdots\!80\)\( p^{75} T^{7} + \)\(64\!\cdots\!47\)\( p^{150} T^{8} - \)\(25\!\cdots\!20\)\( p^{225} T^{9} + \)\(37\!\cdots\!70\)\( p^{300} T^{10} - \)\(79\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 89 | \( 1 + \)\(53\!\cdots\!60\)\( T + \)\(17\!\cdots\!94\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!15\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!00\)\( p^{75} T^{7} + \)\(76\!\cdots\!15\)\( p^{150} T^{8} + \)\(40\!\cdots\!00\)\( p^{225} T^{9} + \)\(17\!\cdots\!94\)\( p^{300} T^{10} + \)\(53\!\cdots\!60\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
| 97 | \( 1 + \)\(74\!\cdots\!80\)\( T + \)\(63\!\cdots\!30\)\( T^{2} + \)\(32\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} + \)\(61\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!40\)\( p^{75} T^{7} + \)\(16\!\cdots\!47\)\( p^{150} T^{8} + \)\(32\!\cdots\!40\)\( p^{225} T^{9} + \)\(63\!\cdots\!30\)\( p^{300} T^{10} + \)\(74\!\cdots\!80\)\( p^{375} T^{11} + p^{450} T^{12} \) | |
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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
−5.19886552913315570098395711479, −4.49488802567913280587814576359, −4.46789405020613708908401386370, −4.39291337610853491641956921440, −4.26442197765479830063498176517, −4.23759162576005352135596586286, −4.04771227342271623544713294680, −3.63137253886059713558126964655, −3.53260251989474910363728890847, −3.46484601400298691448560973926, −3.17500128881955722041128260096, −3.09399782105205743101527898554, −3.01197969896575095959713994522, −2.42995536897018128886757971512, −2.42461968095617423403933912507, −2.31589275250854048492789397217, −2.02486725830125713365423255948, −1.73506701341660733318857023895, −1.71620573632080962516649309151, −1.69377468891130516496515390956, −1.56314875410921696155743011799, −1.04727702741094415896208566905, −0.965956623043206634757132764099, −0.895267073031497054763753729614, −0.801277384034727705860552902398, 0, 0, 0, 0, 0, 0, 0.801277384034727705860552902398, 0.895267073031497054763753729614, 0.965956623043206634757132764099, 1.04727702741094415896208566905, 1.56314875410921696155743011799, 1.69377468891130516496515390956, 1.71620573632080962516649309151, 1.73506701341660733318857023895, 2.02486725830125713365423255948, 2.31589275250854048492789397217, 2.42461968095617423403933912507, 2.42995536897018128886757971512, 3.01197969896575095959713994522, 3.09399782105205743101527898554, 3.17500128881955722041128260096, 3.46484601400298691448560973926, 3.53260251989474910363728890847, 3.63137253886059713558126964655, 4.04771227342271623544713294680, 4.23759162576005352135596586286, 4.26442197765479830063498176517, 4.39291337610853491641956921440, 4.46789405020613708908401386370, 4.49488802567913280587814576359, 5.19886552913315570098395711479
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.