Dirichlet series
| L(s) = 1 | + 2.64e11·2-s + 1.44e18·3-s − 2.11e23·4-s − 2.62e26·5-s + 3.81e29·6-s + 2.70e32·7-s − 8.24e34·8-s − 1.78e37·9-s − 6.93e37·10-s − 1.82e40·11-s − 3.04e41·12-s + 2.38e42·13-s + 7.17e43·14-s − 3.77e44·15-s + 1.19e46·16-s − 3.54e47·17-s − 4.72e48·18-s + 5.60e48·19-s + 5.53e49·20-s + 3.90e50·21-s − 4.83e51·22-s − 4.77e52·23-s − 1.18e53·24-s − 1.34e54·25-s + 6.30e53·26-s − 3.53e55·27-s − 5.71e55·28-s + ⋯ |
| L(s) = 1 | + 0.680·2-s + 0.616·3-s − 1.39·4-s − 0.322·5-s + 0.419·6-s + 0.787·7-s − 1.40·8-s − 3.25·9-s − 0.219·10-s − 1.47·11-s − 0.860·12-s + 0.308·13-s + 0.536·14-s − 0.198·15-s + 0.524·16-s − 1.50·17-s − 2.21·18-s + 0.328·19-s + 0.449·20-s + 0.485·21-s − 1.00·22-s − 1.78·23-s − 0.864·24-s − 2.02·25-s + 0.210·26-s − 2.75·27-s − 1.10·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(78-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+77/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.80231\times 10^{9}\) |
| Root analytic conductor: | \(6.12763\) |
| Motivic weight: | \(77\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 1,\ (\ :[77/2]^{6}),\ 1)\) |
Particular Values
| \(L(39)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{79}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| good | 2 | \( 1 - 1034069895 p^{8} T + 68638101483941228055 p^{12} T^{2} - \)\(14\!\cdots\!95\)\( p^{25} T^{3} + \)\(17\!\cdots\!01\)\( p^{41} T^{4} - \)\(28\!\cdots\!35\)\( p^{64} T^{5} + \)\(99\!\cdots\!95\)\( p^{85} T^{6} - \)\(28\!\cdots\!35\)\( p^{141} T^{7} + \)\(17\!\cdots\!01\)\( p^{195} T^{8} - \)\(14\!\cdots\!95\)\( p^{256} T^{9} + 68638101483941228055 p^{320} T^{10} - 1034069895 p^{393} T^{11} + p^{462} T^{12} \) |
| 3 | \( 1 - 160179041754232120 p^{2} T + \)\(91\!\cdots\!30\)\( p^{7} T^{2} - \)\(14\!\cdots\!80\)\( p^{17} T^{3} + \)\(84\!\cdots\!87\)\( p^{28} T^{4} - \)\(13\!\cdots\!40\)\( p^{40} T^{5} + \)\(27\!\cdots\!20\)\( p^{58} T^{6} - \)\(13\!\cdots\!40\)\( p^{117} T^{7} + \)\(84\!\cdots\!87\)\( p^{182} T^{8} - \)\(14\!\cdots\!80\)\( p^{248} T^{9} + \)\(91\!\cdots\!30\)\( p^{315} T^{10} - 160179041754232120 p^{387} T^{11} + p^{462} T^{12} \) | |
| 5 | \( 1 + \)\(10\!\cdots\!24\)\( p^{2} T + \)\(18\!\cdots\!14\)\( p^{7} T^{2} + \)\(61\!\cdots\!08\)\( p^{14} T^{3} + \)\(33\!\cdots\!71\)\( p^{22} T^{4} + \)\(93\!\cdots\!16\)\( p^{31} T^{5} + \)\(18\!\cdots\!64\)\( p^{42} T^{6} + \)\(93\!\cdots\!16\)\( p^{108} T^{7} + \)\(33\!\cdots\!71\)\( p^{176} T^{8} + \)\(61\!\cdots\!08\)\( p^{245} T^{9} + \)\(18\!\cdots\!14\)\( p^{315} T^{10} + \)\(10\!\cdots\!24\)\( p^{387} T^{11} + p^{462} T^{12} \) | |
| 7 | \( 1 - \)\(38\!\cdots\!00\)\( p T + \)\(45\!\cdots\!50\)\( p^{3} T^{2} - \)\(58\!\cdots\!00\)\( p^{6} T^{3} + \)\(25\!\cdots\!47\)\( p^{12} T^{4} - \)\(11\!\cdots\!00\)\( p^{19} T^{5} + \)\(56\!\cdots\!00\)\( p^{27} T^{6} - \)\(11\!\cdots\!00\)\( p^{96} T^{7} + \)\(25\!\cdots\!47\)\( p^{166} T^{8} - \)\(58\!\cdots\!00\)\( p^{237} T^{9} + \)\(45\!\cdots\!50\)\( p^{311} T^{10} - \)\(38\!\cdots\!00\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 11 | \( 1 + \)\(16\!\cdots\!88\)\( p T + \)\(51\!\cdots\!06\)\( p^{3} T^{2} + \)\(63\!\cdots\!80\)\( p^{6} T^{3} + \)\(98\!\cdots\!45\)\( p^{9} T^{4} + \)\(89\!\cdots\!68\)\( p^{13} T^{5} + \)\(81\!\cdots\!44\)\( p^{18} T^{6} + \)\(89\!\cdots\!68\)\( p^{90} T^{7} + \)\(98\!\cdots\!45\)\( p^{163} T^{8} + \)\(63\!\cdots\!80\)\( p^{237} T^{9} + \)\(51\!\cdots\!06\)\( p^{311} T^{10} + \)\(16\!\cdots\!88\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 13 | \( 1 - \)\(18\!\cdots\!20\)\( p T + \)\(10\!\cdots\!10\)\( p^{3} T^{2} + \)\(44\!\cdots\!40\)\( p^{5} T^{3} + \)\(33\!\cdots\!51\)\( p^{7} T^{4} + \)\(51\!\cdots\!80\)\( p^{10} T^{5} + \)\(25\!\cdots\!80\)\( p^{15} T^{6} + \)\(51\!\cdots\!80\)\( p^{87} T^{7} + \)\(33\!\cdots\!51\)\( p^{161} T^{8} + \)\(44\!\cdots\!40\)\( p^{236} T^{9} + \)\(10\!\cdots\!10\)\( p^{311} T^{10} - \)\(18\!\cdots\!20\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 17 | \( 1 + \)\(20\!\cdots\!20\)\( p T + \)\(79\!\cdots\!10\)\( p^{2} T^{2} + \)\(76\!\cdots\!80\)\( p^{4} T^{3} + \)\(63\!\cdots\!19\)\( p^{7} T^{4} + \)\(17\!\cdots\!40\)\( p^{11} T^{5} + \)\(63\!\cdots\!40\)\( p^{15} T^{6} + \)\(17\!\cdots\!40\)\( p^{88} T^{7} + \)\(63\!\cdots\!19\)\( p^{161} T^{8} + \)\(76\!\cdots\!80\)\( p^{235} T^{9} + \)\(79\!\cdots\!10\)\( p^{310} T^{10} + \)\(20\!\cdots\!20\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 19 | \( 1 - \)\(29\!\cdots\!40\)\( p T + \)\(16\!\cdots\!94\)\( p^{2} T^{2} - \)\(21\!\cdots\!00\)\( p^{5} T^{3} + \)\(62\!\cdots\!15\)\( p^{6} T^{4} - \)\(15\!\cdots\!00\)\( p^{8} T^{5} + \)\(14\!\cdots\!80\)\( p^{10} T^{6} - \)\(15\!\cdots\!00\)\( p^{85} T^{7} + \)\(62\!\cdots\!15\)\( p^{160} T^{8} - \)\(21\!\cdots\!00\)\( p^{236} T^{9} + \)\(16\!\cdots\!94\)\( p^{310} T^{10} - \)\(29\!\cdots\!40\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 23 | \( 1 + \)\(47\!\cdots\!60\)\( T + \)\(13\!\cdots\!10\)\( p T^{2} + \)\(90\!\cdots\!40\)\( p^{3} T^{3} + \)\(67\!\cdots\!89\)\( p^{5} T^{4} + \)\(16\!\cdots\!80\)\( p^{8} T^{5} + \)\(39\!\cdots\!20\)\( p^{11} T^{6} + \)\(16\!\cdots\!80\)\( p^{85} T^{7} + \)\(67\!\cdots\!89\)\( p^{159} T^{8} + \)\(90\!\cdots\!40\)\( p^{234} T^{9} + \)\(13\!\cdots\!10\)\( p^{309} T^{10} + \)\(47\!\cdots\!60\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 29 | \( 1 - \)\(16\!\cdots\!40\)\( T + \)\(51\!\cdots\!26\)\( p T^{2} - \)\(41\!\cdots\!00\)\( p^{2} T^{3} + \)\(15\!\cdots\!15\)\( p^{4} T^{4} - \)\(10\!\cdots\!00\)\( p^{6} T^{5} + \)\(11\!\cdots\!80\)\( p^{8} T^{6} - \)\(10\!\cdots\!00\)\( p^{83} T^{7} + \)\(15\!\cdots\!15\)\( p^{158} T^{8} - \)\(41\!\cdots\!00\)\( p^{233} T^{9} + \)\(51\!\cdots\!26\)\( p^{309} T^{10} - \)\(16\!\cdots\!40\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 31 | \( 1 - \)\(17\!\cdots\!72\)\( T + \)\(73\!\cdots\!46\)\( p T^{2} - \)\(52\!\cdots\!20\)\( p^{2} T^{3} + \)\(93\!\cdots\!45\)\( p^{3} T^{4} - \)\(21\!\cdots\!92\)\( p^{5} T^{5} + \)\(83\!\cdots\!04\)\( p^{7} T^{6} - \)\(21\!\cdots\!92\)\( p^{82} T^{7} + \)\(93\!\cdots\!45\)\( p^{157} T^{8} - \)\(52\!\cdots\!20\)\( p^{233} T^{9} + \)\(73\!\cdots\!46\)\( p^{309} T^{10} - \)\(17\!\cdots\!72\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 37 | \( 1 + \)\(20\!\cdots\!60\)\( p T + \)\(36\!\cdots\!30\)\( p^{2} T^{2} + \)\(42\!\cdots\!80\)\( p^{3} T^{3} + \)\(11\!\cdots\!31\)\( p^{5} T^{4} + \)\(25\!\cdots\!20\)\( p^{7} T^{5} + \)\(49\!\cdots\!80\)\( p^{9} T^{6} + \)\(25\!\cdots\!20\)\( p^{84} T^{7} + \)\(11\!\cdots\!31\)\( p^{159} T^{8} + \)\(42\!\cdots\!80\)\( p^{234} T^{9} + \)\(36\!\cdots\!30\)\( p^{310} T^{10} + \)\(20\!\cdots\!60\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 41 | \( 1 + \)\(62\!\cdots\!08\)\( T + \)\(22\!\cdots\!46\)\( T^{2} + \)\(13\!\cdots\!80\)\( p T^{3} + \)\(64\!\cdots\!95\)\( p^{2} T^{4} + \)\(25\!\cdots\!48\)\( p^{3} T^{5} + \)\(82\!\cdots\!44\)\( p^{4} T^{6} + \)\(25\!\cdots\!48\)\( p^{80} T^{7} + \)\(64\!\cdots\!95\)\( p^{156} T^{8} + \)\(13\!\cdots\!80\)\( p^{232} T^{9} + \)\(22\!\cdots\!46\)\( p^{308} T^{10} + \)\(62\!\cdots\!08\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 43 | \( 1 + \)\(26\!\cdots\!00\)\( T + \)\(53\!\cdots\!50\)\( T^{2} + \)\(78\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!29\)\( p T^{4} + \)\(50\!\cdots\!00\)\( p^{2} T^{5} + \)\(98\!\cdots\!00\)\( p^{3} T^{6} + \)\(50\!\cdots\!00\)\( p^{79} T^{7} + \)\(21\!\cdots\!29\)\( p^{155} T^{8} + \)\(78\!\cdots\!00\)\( p^{231} T^{9} + \)\(53\!\cdots\!50\)\( p^{308} T^{10} + \)\(26\!\cdots\!00\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 47 | \( 1 - \)\(10\!\cdots\!40\)\( T + \)\(19\!\cdots\!10\)\( T^{2} - \)\(99\!\cdots\!20\)\( p^{2} T^{3} + \)\(84\!\cdots\!23\)\( p^{2} T^{4} - \)\(19\!\cdots\!40\)\( p^{3} T^{5} + \)\(25\!\cdots\!80\)\( p^{4} T^{6} - \)\(19\!\cdots\!40\)\( p^{80} T^{7} + \)\(84\!\cdots\!23\)\( p^{156} T^{8} - \)\(99\!\cdots\!20\)\( p^{233} T^{9} + \)\(19\!\cdots\!10\)\( p^{308} T^{10} - \)\(10\!\cdots\!40\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 53 | \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(28\!\cdots\!70\)\( p T^{2} + \)\(38\!\cdots\!40\)\( p^{2} T^{3} + \)\(43\!\cdots\!91\)\( p^{3} T^{4} - \)\(31\!\cdots\!40\)\( p^{4} T^{5} + \)\(25\!\cdots\!60\)\( p^{5} T^{6} - \)\(31\!\cdots\!40\)\( p^{81} T^{7} + \)\(43\!\cdots\!91\)\( p^{157} T^{8} + \)\(38\!\cdots\!40\)\( p^{233} T^{9} + \)\(28\!\cdots\!70\)\( p^{309} T^{10} + \)\(13\!\cdots\!20\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 59 | \( 1 + \)\(31\!\cdots\!20\)\( T + \)\(24\!\cdots\!46\)\( p T^{2} + \)\(88\!\cdots\!00\)\( p^{2} T^{3} + \)\(38\!\cdots\!85\)\( p^{3} T^{4} + \)\(10\!\cdots\!00\)\( p^{4} T^{5} + \)\(33\!\cdots\!20\)\( p^{5} T^{6} + \)\(10\!\cdots\!00\)\( p^{81} T^{7} + \)\(38\!\cdots\!85\)\( p^{157} T^{8} + \)\(88\!\cdots\!00\)\( p^{233} T^{9} + \)\(24\!\cdots\!46\)\( p^{309} T^{10} + \)\(31\!\cdots\!20\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 61 | \( 1 - \)\(12\!\cdots\!32\)\( T + \)\(35\!\cdots\!26\)\( p T^{2} - \)\(46\!\cdots\!20\)\( p^{2} T^{3} + \)\(74\!\cdots\!95\)\( p^{3} T^{4} - \)\(71\!\cdots\!12\)\( p^{4} T^{5} + \)\(79\!\cdots\!64\)\( p^{5} T^{6} - \)\(71\!\cdots\!12\)\( p^{81} T^{7} + \)\(74\!\cdots\!95\)\( p^{157} T^{8} - \)\(46\!\cdots\!20\)\( p^{233} T^{9} + \)\(35\!\cdots\!26\)\( p^{309} T^{10} - \)\(12\!\cdots\!32\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 67 | \( 1 + \)\(20\!\cdots\!40\)\( T + \)\(24\!\cdots\!70\)\( p T^{2} + \)\(73\!\cdots\!20\)\( p^{2} T^{3} + \)\(44\!\cdots\!49\)\( p^{3} T^{4} + \)\(11\!\cdots\!20\)\( p^{4} T^{5} + \)\(50\!\cdots\!60\)\( p^{5} T^{6} + \)\(11\!\cdots\!20\)\( p^{81} T^{7} + \)\(44\!\cdots\!49\)\( p^{157} T^{8} + \)\(73\!\cdots\!20\)\( p^{233} T^{9} + \)\(24\!\cdots\!70\)\( p^{309} T^{10} + \)\(20\!\cdots\!40\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 71 | \( 1 + \)\(11\!\cdots\!88\)\( p T + \)\(26\!\cdots\!66\)\( p^{2} T^{2} + \)\(18\!\cdots\!80\)\( p^{3} T^{3} + \)\(34\!\cdots\!95\)\( p^{4} T^{4} + \)\(14\!\cdots\!08\)\( p^{5} T^{5} + \)\(28\!\cdots\!24\)\( p^{6} T^{6} + \)\(14\!\cdots\!08\)\( p^{82} T^{7} + \)\(34\!\cdots\!95\)\( p^{158} T^{8} + \)\(18\!\cdots\!80\)\( p^{234} T^{9} + \)\(26\!\cdots\!66\)\( p^{310} T^{10} + \)\(11\!\cdots\!88\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 73 | \( 1 - \)\(43\!\cdots\!80\)\( p T + \)\(13\!\cdots\!70\)\( p^{2} T^{2} + \)\(24\!\cdots\!40\)\( p^{3} T^{3} + \)\(10\!\cdots\!47\)\( p^{4} T^{4} + \)\(10\!\cdots\!60\)\( p^{5} T^{5} + \)\(70\!\cdots\!60\)\( p^{6} T^{6} + \)\(10\!\cdots\!60\)\( p^{82} T^{7} + \)\(10\!\cdots\!47\)\( p^{158} T^{8} + \)\(24\!\cdots\!40\)\( p^{234} T^{9} + \)\(13\!\cdots\!70\)\( p^{310} T^{10} - \)\(43\!\cdots\!80\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 79 | \( 1 + \)\(44\!\cdots\!60\)\( T + \)\(61\!\cdots\!54\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!15\)\( T^{4} + \)\(40\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} + \)\(40\!\cdots\!00\)\( p^{77} T^{7} + \)\(16\!\cdots\!15\)\( p^{154} T^{8} + \)\(19\!\cdots\!00\)\( p^{231} T^{9} + \)\(61\!\cdots\!54\)\( p^{308} T^{10} + \)\(44\!\cdots\!60\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 83 | \( 1 + \)\(14\!\cdots\!80\)\( T + \)\(22\!\cdots\!90\)\( T^{2} + \)\(31\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!87\)\( T^{4} + \)\(27\!\cdots\!40\)\( T^{5} + \)\(24\!\cdots\!20\)\( T^{6} + \)\(27\!\cdots\!40\)\( p^{77} T^{7} + \)\(30\!\cdots\!87\)\( p^{154} T^{8} + \)\(31\!\cdots\!40\)\( p^{231} T^{9} + \)\(22\!\cdots\!90\)\( p^{308} T^{10} + \)\(14\!\cdots\!80\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 89 | \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(36\!\cdots\!74\)\( T^{2} - \)\(34\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!15\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!00\)\( p^{77} T^{7} + \)\(74\!\cdots\!15\)\( p^{154} T^{8} - \)\(34\!\cdots\!00\)\( p^{231} T^{9} + \)\(36\!\cdots\!74\)\( p^{308} T^{10} - \)\(24\!\cdots\!20\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 97 | \( 1 + \)\(12\!\cdots\!60\)\( T + \)\(48\!\cdots\!10\)\( T^{2} + \)\(36\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!07\)\( T^{4} + \)\(49\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!80\)\( T^{6} + \)\(49\!\cdots\!80\)\( p^{77} T^{7} + \)\(10\!\cdots\!07\)\( p^{154} T^{8} + \)\(36\!\cdots\!20\)\( p^{231} T^{9} + \)\(48\!\cdots\!10\)\( p^{308} T^{10} + \)\(12\!\cdots\!60\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
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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
−8.098994977694216503554746150915, −7.73359636451078555992886593182, −7.66534875364885692987863943623, −6.99689937854500642367455892411, −6.59979783550000133316732570075, −6.26793333844939544423092155334, −6.26405123041996032464585117561, −5.61672782074057470543426505704, −5.39114441230915839220913556835, −5.37208176044697332874969272562, −5.06135861140998656342037062409, −4.75929443148309090120962963598, −4.70077324727644321587584438177, −4.17598447306215646840419955191, −3.59903709682804289731794552199, −3.55153081507190330367967364688, −3.48460458608087259110157283530, −3.28996470810543097775428955450, −2.72130235567142653526662454888, −2.50795140872845410422367768551, −2.36854966842760420614828607236, −1.80611216187130323016099483013, −1.68083763095949248529949656486, −1.59539358650695155920198470023, −1.01967603610485487474948452430, 0, 0, 0, 0, 0, 0, 1.01967603610485487474948452430, 1.59539358650695155920198470023, 1.68083763095949248529949656486, 1.80611216187130323016099483013, 2.36854966842760420614828607236, 2.50795140872845410422367768551, 2.72130235567142653526662454888, 3.28996470810543097775428955450, 3.48460458608087259110157283530, 3.55153081507190330367967364688, 3.59903709682804289731794552199, 4.17598447306215646840419955191, 4.70077324727644321587584438177, 4.75929443148309090120962963598, 5.06135861140998656342037062409, 5.37208176044697332874969272562, 5.39114441230915839220913556835, 5.61672782074057470543426505704, 6.26405123041996032464585117561, 6.26793333844939544423092155334, 6.59979783550000133316732570075, 6.99689937854500642367455892411, 7.66534875364885692987863943623, 7.73359636451078555992886593182, 8.098994977694216503554746150915
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.