Dirichlet series
| L(s) = 1 | − 1.60e10·2-s + 1.94e18·3-s − 1.03e24·4-s + 6.09e27·5-s − 3.12e28·6-s − 2.04e32·7-s + 2.00e35·8-s − 9.66e37·9-s − 9.80e37·10-s + 3.26e41·11-s − 2.01e42·12-s − 2.49e44·13-s + 3.28e42·14-s + 1.18e46·15-s + 1.31e47·16-s + 8.29e47·17-s + 1.55e48·18-s − 5.07e49·19-s − 6.32e51·20-s − 3.96e50·21-s − 5.24e51·22-s − 1.04e54·23-s + 3.89e53·24-s − 2.56e55·25-s + 4.01e54·26-s − 7.13e55·27-s + 2.11e56·28-s + ⋯ |
| L(s) = 1 | − 0.0206·2-s + 0.276·3-s − 1.71·4-s + 1.49·5-s − 0.00572·6-s − 0.0848·7-s + 0.427·8-s − 1.96·9-s − 0.0309·10-s + 2.39·11-s − 0.475·12-s − 2.48·13-s + 0.00175·14-s + 0.414·15-s + 0.360·16-s + 0.207·17-s + 0.0405·18-s − 0.156·19-s − 2.57·20-s − 0.0234·21-s − 0.0494·22-s − 1.70·23-s + 0.118·24-s − 1.55·25-s + 0.0514·26-s − 0.206·27-s + 0.145·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(80-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+79/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.81193\times 10^{9}\) |
| Root analytic conductor: | \(6.28678\) |
| Motivic weight: | \(79\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 1,\ (\ :[79/2]^{6}),\ 1)\) |
Particular Values
| \(L(40)\) | \(\approx\) | \(5.500276106\) |
| \(L(\frac12)\) | \(\approx\) | \(5.500276106\) |
| \(L(\frac{81}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| good | 2 | \( 1 + 2010822165 p^{3} T + \)\(10\!\cdots\!05\)\( p^{10} T^{2} - \)\(39\!\cdots\!35\)\( p^{22} T^{3} + \)\(68\!\cdots\!31\)\( p^{37} T^{4} - \)\(51\!\cdots\!15\)\( p^{56} T^{5} + \)\(11\!\cdots\!95\)\( p^{79} T^{6} - \)\(51\!\cdots\!15\)\( p^{135} T^{7} + \)\(68\!\cdots\!31\)\( p^{195} T^{8} - \)\(39\!\cdots\!35\)\( p^{259} T^{9} + \)\(10\!\cdots\!05\)\( p^{326} T^{10} + 2010822165 p^{398} T^{11} + p^{474} T^{12} \) |
| 3 | \( 1 - 71932404751154840 p^{3} T + \)\(56\!\cdots\!70\)\( p^{11} T^{2} - \)\(29\!\cdots\!20\)\( p^{21} T^{3} + \)\(13\!\cdots\!29\)\( p^{33} T^{4} - \)\(97\!\cdots\!20\)\( p^{47} T^{5} + \)\(38\!\cdots\!60\)\( p^{63} T^{6} - \)\(97\!\cdots\!20\)\( p^{126} T^{7} + \)\(13\!\cdots\!29\)\( p^{191} T^{8} - \)\(29\!\cdots\!20\)\( p^{258} T^{9} + \)\(56\!\cdots\!70\)\( p^{327} T^{10} - 71932404751154840 p^{398} T^{11} + p^{474} T^{12} \) | |
| 5 | \( 1 - \)\(12\!\cdots\!08\)\( p T + \)\(20\!\cdots\!58\)\( p^{5} T^{2} - \)\(28\!\cdots\!68\)\( p^{10} T^{3} + \)\(52\!\cdots\!43\)\( p^{18} T^{4} - \)\(96\!\cdots\!16\)\( p^{27} T^{5} + \)\(54\!\cdots\!52\)\( p^{37} T^{6} - \)\(96\!\cdots\!16\)\( p^{106} T^{7} + \)\(52\!\cdots\!43\)\( p^{176} T^{8} - \)\(28\!\cdots\!68\)\( p^{247} T^{9} + \)\(20\!\cdots\!58\)\( p^{321} T^{10} - \)\(12\!\cdots\!08\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 7 | \( 1 + \)\(41\!\cdots\!00\)\( p^{2} T + \)\(92\!\cdots\!50\)\( p^{5} T^{2} + \)\(10\!\cdots\!00\)\( p^{10} T^{3} + \)\(57\!\cdots\!21\)\( p^{17} T^{4} - \)\(78\!\cdots\!00\)\( p^{25} T^{5} + \)\(16\!\cdots\!00\)\( p^{34} T^{6} - \)\(78\!\cdots\!00\)\( p^{104} T^{7} + \)\(57\!\cdots\!21\)\( p^{175} T^{8} + \)\(10\!\cdots\!00\)\( p^{247} T^{9} + \)\(92\!\cdots\!50\)\( p^{321} T^{10} + \)\(41\!\cdots\!00\)\( p^{397} T^{11} + p^{474} T^{12} \) | |
| 11 | \( 1 - \)\(29\!\cdots\!32\)\( p T + \)\(66\!\cdots\!26\)\( p^{3} T^{2} - \)\(92\!\cdots\!20\)\( p^{6} T^{3} + \)\(11\!\cdots\!95\)\( p^{10} T^{4} - \)\(12\!\cdots\!12\)\( p^{14} T^{5} + \)\(11\!\cdots\!44\)\( p^{19} T^{6} - \)\(12\!\cdots\!12\)\( p^{93} T^{7} + \)\(11\!\cdots\!95\)\( p^{168} T^{8} - \)\(92\!\cdots\!20\)\( p^{243} T^{9} + \)\(66\!\cdots\!26\)\( p^{319} T^{10} - \)\(29\!\cdots\!32\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 13 | \( 1 + \)\(19\!\cdots\!80\)\( p T + \)\(20\!\cdots\!90\)\( p^{3} T^{2} + \)\(13\!\cdots\!20\)\( p^{6} T^{3} + \)\(77\!\cdots\!19\)\( p^{9} T^{4} + \)\(41\!\cdots\!20\)\( p^{12} T^{5} + \)\(15\!\cdots\!40\)\( p^{16} T^{6} + \)\(41\!\cdots\!20\)\( p^{91} T^{7} + \)\(77\!\cdots\!19\)\( p^{167} T^{8} + \)\(13\!\cdots\!20\)\( p^{243} T^{9} + \)\(20\!\cdots\!90\)\( p^{319} T^{10} + \)\(19\!\cdots\!80\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 17 | \( 1 - \)\(82\!\cdots\!40\)\( T + \)\(26\!\cdots\!30\)\( p T^{2} - \)\(19\!\cdots\!40\)\( p^{3} T^{3} + \)\(87\!\cdots\!11\)\( p^{5} T^{4} - \)\(32\!\cdots\!20\)\( p^{8} T^{5} + \)\(44\!\cdots\!80\)\( p^{12} T^{6} - \)\(32\!\cdots\!20\)\( p^{87} T^{7} + \)\(87\!\cdots\!11\)\( p^{163} T^{8} - \)\(19\!\cdots\!40\)\( p^{240} T^{9} + \)\(26\!\cdots\!30\)\( p^{317} T^{10} - \)\(82\!\cdots\!40\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 19 | \( 1 + \)\(26\!\cdots\!40\)\( p T + \)\(59\!\cdots\!86\)\( p^{3} T^{2} + \)\(23\!\cdots\!00\)\( p^{5} T^{3} + \)\(43\!\cdots\!15\)\( p^{8} T^{4} + \)\(12\!\cdots\!00\)\( p^{11} T^{5} + \)\(11\!\cdots\!80\)\( p^{14} T^{6} + \)\(12\!\cdots\!00\)\( p^{90} T^{7} + \)\(43\!\cdots\!15\)\( p^{166} T^{8} + \)\(23\!\cdots\!00\)\( p^{242} T^{9} + \)\(59\!\cdots\!86\)\( p^{319} T^{10} + \)\(26\!\cdots\!40\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 23 | \( 1 + \)\(45\!\cdots\!20\)\( p T + \)\(34\!\cdots\!30\)\( p^{2} T^{2} + \)\(12\!\cdots\!60\)\( p^{3} T^{3} + \)\(23\!\cdots\!49\)\( p^{5} T^{4} + \)\(12\!\cdots\!80\)\( p^{8} T^{5} + \)\(77\!\cdots\!80\)\( p^{11} T^{6} + \)\(12\!\cdots\!80\)\( p^{87} T^{7} + \)\(23\!\cdots\!49\)\( p^{163} T^{8} + \)\(12\!\cdots\!60\)\( p^{240} T^{9} + \)\(34\!\cdots\!30\)\( p^{318} T^{10} + \)\(45\!\cdots\!20\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 29 | \( 1 + \)\(71\!\cdots\!40\)\( T + \)\(76\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!00\)\( p T^{3} + \)\(48\!\cdots\!15\)\( p^{2} T^{4} + \)\(26\!\cdots\!00\)\( p^{4} T^{5} + \)\(21\!\cdots\!80\)\( p^{6} T^{6} + \)\(26\!\cdots\!00\)\( p^{83} T^{7} + \)\(48\!\cdots\!15\)\( p^{160} T^{8} + \)\(15\!\cdots\!00\)\( p^{238} T^{9} + \)\(76\!\cdots\!14\)\( p^{316} T^{10} + \)\(71\!\cdots\!40\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 31 | \( 1 - \)\(13\!\cdots\!32\)\( T + \)\(12\!\cdots\!06\)\( p T^{2} - \)\(41\!\cdots\!20\)\( p^{2} T^{3} + \)\(64\!\cdots\!95\)\( p^{4} T^{4} - \)\(56\!\cdots\!32\)\( p^{6} T^{5} + \)\(60\!\cdots\!04\)\( p^{8} T^{6} - \)\(56\!\cdots\!32\)\( p^{85} T^{7} + \)\(64\!\cdots\!95\)\( p^{162} T^{8} - \)\(41\!\cdots\!20\)\( p^{239} T^{9} + \)\(12\!\cdots\!06\)\( p^{317} T^{10} - \)\(13\!\cdots\!32\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 37 | \( 1 - \)\(54\!\cdots\!60\)\( p T + \)\(37\!\cdots\!70\)\( p^{2} T^{2} - \)\(13\!\cdots\!20\)\( p^{3} T^{3} + \)\(14\!\cdots\!91\)\( p^{5} T^{4} - \)\(10\!\cdots\!20\)\( p^{7} T^{5} + \)\(80\!\cdots\!20\)\( p^{9} T^{6} - \)\(10\!\cdots\!20\)\( p^{86} T^{7} + \)\(14\!\cdots\!91\)\( p^{163} T^{8} - \)\(13\!\cdots\!20\)\( p^{240} T^{9} + \)\(37\!\cdots\!70\)\( p^{318} T^{10} - \)\(54\!\cdots\!60\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 41 | \( 1 - \)\(19\!\cdots\!72\)\( T + \)\(25\!\cdots\!26\)\( T^{2} - \)\(24\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!95\)\( p T^{4} - \)\(72\!\cdots\!32\)\( p^{2} T^{5} + \)\(97\!\cdots\!64\)\( p^{3} T^{6} - \)\(72\!\cdots\!32\)\( p^{81} T^{7} + \)\(46\!\cdots\!95\)\( p^{159} T^{8} - \)\(24\!\cdots\!20\)\( p^{237} T^{9} + \)\(25\!\cdots\!26\)\( p^{316} T^{10} - \)\(19\!\cdots\!72\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 43 | \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(79\!\cdots\!50\)\( T^{2} - \)\(10\!\cdots\!00\)\( p T^{3} + \)\(12\!\cdots\!03\)\( p^{2} T^{4} - \)\(11\!\cdots\!00\)\( p^{3} T^{5} + \)\(96\!\cdots\!00\)\( p^{4} T^{6} - \)\(11\!\cdots\!00\)\( p^{82} T^{7} + \)\(12\!\cdots\!03\)\( p^{160} T^{8} - \)\(10\!\cdots\!00\)\( p^{238} T^{9} + \)\(79\!\cdots\!50\)\( p^{316} T^{10} - \)\(10\!\cdots\!00\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 47 | \( 1 - \)\(16\!\cdots\!80\)\( p T + \)\(65\!\cdots\!90\)\( T^{2} - \)\(92\!\cdots\!40\)\( p T^{3} + \)\(85\!\cdots\!63\)\( p^{2} T^{4} - \)\(99\!\cdots\!60\)\( p^{3} T^{5} + \)\(62\!\cdots\!20\)\( p^{4} T^{6} - \)\(99\!\cdots\!60\)\( p^{82} T^{7} + \)\(85\!\cdots\!63\)\( p^{160} T^{8} - \)\(92\!\cdots\!40\)\( p^{238} T^{9} + \)\(65\!\cdots\!90\)\( p^{316} T^{10} - \)\(16\!\cdots\!80\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 53 | \( 1 + \)\(31\!\cdots\!20\)\( T + \)\(95\!\cdots\!90\)\( T^{2} + \)\(32\!\cdots\!80\)\( p T^{3} + \)\(11\!\cdots\!63\)\( p^{2} T^{4} + \)\(32\!\cdots\!80\)\( p^{3} T^{5} + \)\(91\!\cdots\!20\)\( p^{4} T^{6} + \)\(32\!\cdots\!80\)\( p^{82} T^{7} + \)\(11\!\cdots\!63\)\( p^{160} T^{8} + \)\(32\!\cdots\!80\)\( p^{238} T^{9} + \)\(95\!\cdots\!90\)\( p^{316} T^{10} + \)\(31\!\cdots\!20\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 59 | \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(72\!\cdots\!26\)\( p T^{2} - \)\(15\!\cdots\!00\)\( p^{2} T^{3} + \)\(29\!\cdots\!85\)\( p^{3} T^{4} - \)\(48\!\cdots\!00\)\( p^{4} T^{5} + \)\(73\!\cdots\!20\)\( p^{5} T^{6} - \)\(48\!\cdots\!00\)\( p^{83} T^{7} + \)\(29\!\cdots\!85\)\( p^{161} T^{8} - \)\(15\!\cdots\!00\)\( p^{239} T^{9} + \)\(72\!\cdots\!26\)\( p^{317} T^{10} - \)\(24\!\cdots\!20\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 61 | \( 1 - \)\(80\!\cdots\!52\)\( T + \)\(11\!\cdots\!46\)\( p T^{2} - \)\(83\!\cdots\!20\)\( p^{2} T^{3} + \)\(68\!\cdots\!95\)\( p^{3} T^{4} - \)\(35\!\cdots\!12\)\( p^{4} T^{5} + \)\(23\!\cdots\!04\)\( p^{5} T^{6} - \)\(35\!\cdots\!12\)\( p^{83} T^{7} + \)\(68\!\cdots\!95\)\( p^{161} T^{8} - \)\(83\!\cdots\!20\)\( p^{239} T^{9} + \)\(11\!\cdots\!46\)\( p^{317} T^{10} - \)\(80\!\cdots\!52\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 67 | \( 1 + \)\(27\!\cdots\!80\)\( p T + \)\(18\!\cdots\!90\)\( p^{2} T^{2} + \)\(42\!\cdots\!60\)\( p^{3} T^{3} + \)\(16\!\cdots\!87\)\( p^{4} T^{4} + \)\(29\!\cdots\!40\)\( p^{5} T^{5} + \)\(82\!\cdots\!20\)\( p^{6} T^{6} + \)\(29\!\cdots\!40\)\( p^{84} T^{7} + \)\(16\!\cdots\!87\)\( p^{162} T^{8} + \)\(42\!\cdots\!60\)\( p^{240} T^{9} + \)\(18\!\cdots\!90\)\( p^{318} T^{10} + \)\(27\!\cdots\!80\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 71 | \( 1 - \)\(54\!\cdots\!52\)\( p T + \)\(29\!\cdots\!06\)\( p^{2} T^{2} - \)\(95\!\cdots\!20\)\( p^{3} T^{3} + \)\(29\!\cdots\!95\)\( p^{4} T^{4} - \)\(66\!\cdots\!92\)\( p^{5} T^{5} + \)\(14\!\cdots\!04\)\( p^{6} T^{6} - \)\(66\!\cdots\!92\)\( p^{84} T^{7} + \)\(29\!\cdots\!95\)\( p^{162} T^{8} - \)\(95\!\cdots\!20\)\( p^{240} T^{9} + \)\(29\!\cdots\!06\)\( p^{318} T^{10} - \)\(54\!\cdots\!52\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 73 | \( 1 - \)\(83\!\cdots\!80\)\( p T + \)\(13\!\cdots\!30\)\( p^{2} T^{2} - \)\(78\!\cdots\!40\)\( p^{3} T^{3} + \)\(67\!\cdots\!27\)\( p^{4} T^{4} - \)\(33\!\cdots\!40\)\( p^{5} T^{5} + \)\(22\!\cdots\!40\)\( p^{6} T^{6} - \)\(33\!\cdots\!40\)\( p^{84} T^{7} + \)\(67\!\cdots\!27\)\( p^{162} T^{8} - \)\(78\!\cdots\!40\)\( p^{240} T^{9} + \)\(13\!\cdots\!30\)\( p^{318} T^{10} - \)\(83\!\cdots\!80\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 79 | \( 1 + \)\(18\!\cdots\!40\)\( T + \)\(10\!\cdots\!14\)\( T^{2} + \)\(13\!\cdots\!00\)\( p T^{3} + \)\(64\!\cdots\!15\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(78\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!00\)\( p^{79} T^{7} + \)\(64\!\cdots\!15\)\( p^{158} T^{8} + \)\(13\!\cdots\!00\)\( p^{238} T^{9} + \)\(10\!\cdots\!14\)\( p^{316} T^{10} + \)\(18\!\cdots\!40\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 83 | \( 1 - \)\(14\!\cdots\!20\)\( T + \)\(28\!\cdots\!10\)\( T^{2} - \)\(27\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!27\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(21\!\cdots\!60\)\( p^{79} T^{7} + \)\(30\!\cdots\!27\)\( p^{158} T^{8} - \)\(27\!\cdots\!40\)\( p^{237} T^{9} + \)\(28\!\cdots\!10\)\( p^{316} T^{10} - \)\(14\!\cdots\!20\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 89 | \( 1 - \)\(88\!\cdots\!80\)\( T + \)\(39\!\cdots\!54\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!15\)\( T^{4} + \)\(43\!\cdots\!00\)\( T^{5} + \)\(51\!\cdots\!80\)\( T^{6} + \)\(43\!\cdots\!00\)\( p^{79} T^{7} + \)\(55\!\cdots\!15\)\( p^{158} T^{8} - \)\(14\!\cdots\!00\)\( p^{237} T^{9} + \)\(39\!\cdots\!54\)\( p^{316} T^{10} - \)\(88\!\cdots\!80\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 97 | \( 1 - \)\(20\!\cdots\!60\)\( T + \)\(21\!\cdots\!90\)\( T^{2} - \)\(53\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!67\)\( T^{4} - \)\(73\!\cdots\!80\)\( T^{5} + \)\(38\!\cdots\!20\)\( T^{6} - \)\(73\!\cdots\!80\)\( p^{79} T^{7} + \)\(38\!\cdots\!67\)\( p^{158} T^{8} - \)\(53\!\cdots\!80\)\( p^{237} T^{9} + \)\(21\!\cdots\!90\)\( p^{316} T^{10} - \)\(20\!\cdots\!60\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
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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
−6.61145025976504718644567543284, −6.58509434788769017196571620612, −6.48436546906874165710182175042, −5.94146884513462250177598342614, −5.71420316279497845240420256637, −5.58665963502690916884971894517, −5.44439330053608455545275736723, −4.98732340477794019916006972057, −4.43494186680390215756685092359, −4.35580474518380941313534576081, −4.15203797206811714605804558296, −4.13560599012070053392534113422, −3.74447914094962328498607883221, −3.17174258368265871893196000686, −2.94522297768938828124858349326, −2.63180320104243486760473805870, −2.27486611006901282937269064041, −2.02321218593275360637280896813, −1.99966424155203966989268386725, −1.91880601465280198807570261067, −1.13148549752498362450558244495, −0.72084184862080328275728459102, −0.63944864586061619364386458998, −0.59352203405527027543619141746, −0.25326550789926601380635987089, 0.25326550789926601380635987089, 0.59352203405527027543619141746, 0.63944864586061619364386458998, 0.72084184862080328275728459102, 1.13148549752498362450558244495, 1.91880601465280198807570261067, 1.99966424155203966989268386725, 2.02321218593275360637280896813, 2.27486611006901282937269064041, 2.63180320104243486760473805870, 2.94522297768938828124858349326, 3.17174258368265871893196000686, 3.74447914094962328498607883221, 4.13560599012070053392534113422, 4.15203797206811714605804558296, 4.35580474518380941313534576081, 4.43494186680390215756685092359, 4.98732340477794019916006972057, 5.44439330053608455545275736723, 5.58665963502690916884971894517, 5.71420316279497845240420256637, 5.94146884513462250177598342614, 6.48436546906874165710182175042, 6.58509434788769017196571620612, 6.61145025976504718644567543284
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.