Dirichlet series
| L(s) = 1 | − 8.47e11·2-s − 8.10e18·3-s + 1.65e23·4-s − 1.59e27·5-s + 6.87e30·6-s − 4.60e32·7-s + 5.52e34·8-s + 3.83e37·9-s + 1.35e39·10-s − 3.12e39·11-s − 1.34e42·12-s − 1.07e43·13-s + 3.89e44·14-s + 1.29e46·15-s − 2.72e46·16-s + 2.05e47·17-s − 3.24e49·18-s − 1.86e49·19-s − 2.64e50·20-s + 3.72e51·21-s + 2.64e51·22-s − 1.01e52·23-s − 4.47e53·24-s − 1.51e53·25-s + 9.11e54·26-s − 1.38e56·27-s − 7.62e55·28-s + ⋯ |
| L(s) = 1 | − 2.18·2-s − 3.46·3-s + 1.09·4-s − 1.96·5-s + 7.55·6-s − 1.33·7-s + 0.940·8-s + 7·9-s + 4.27·10-s − 0.251·11-s − 3.80·12-s − 1.39·13-s + 2.91·14-s + 6.79·15-s − 1.19·16-s + 0.869·17-s − 15.2·18-s − 1.09·19-s − 2.15·20-s + 4.63·21-s + 0.549·22-s − 0.381·23-s − 3.25·24-s − 0.228·25-s + 3.04·26-s − 10.7·27-s − 1.46·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(78-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+77/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(729\) = \(3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.04288\times 10^{12}\) |
| Root analytic conductor: | \(10.6133\) |
| Motivic weight: | \(77\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 729,\ (\ :[77/2]^{6}),\ 1)\) |
Particular Values
| \(L(39)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{79}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{38} T )^{6} \) |
| good | 2 | \( 1 + 211921238547 p^{2} T + \)\(21\!\cdots\!69\)\( p^{8} T^{2} + \)\(52\!\cdots\!63\)\( p^{19} T^{3} + \)\(13\!\cdots\!91\)\( p^{33} T^{4} + \)\(21\!\cdots\!07\)\( p^{51} T^{5} + \)\(18\!\cdots\!67\)\( p^{73} T^{6} + \)\(21\!\cdots\!07\)\( p^{128} T^{7} + \)\(13\!\cdots\!91\)\( p^{187} T^{8} + \)\(52\!\cdots\!63\)\( p^{250} T^{9} + \)\(21\!\cdots\!69\)\( p^{316} T^{10} + 211921238547 p^{387} T^{11} + p^{462} T^{12} \) |
| 5 | \( 1 + \)\(31\!\cdots\!68\)\( p T + \)\(21\!\cdots\!82\)\( p^{3} T^{2} + \)\(42\!\cdots\!36\)\( p^{7} T^{3} + \)\(64\!\cdots\!07\)\( p^{14} T^{4} + \)\(15\!\cdots\!16\)\( p^{22} T^{5} + \)\(70\!\cdots\!36\)\( p^{31} T^{6} + \)\(15\!\cdots\!16\)\( p^{99} T^{7} + \)\(64\!\cdots\!07\)\( p^{168} T^{8} + \)\(42\!\cdots\!36\)\( p^{238} T^{9} + \)\(21\!\cdots\!82\)\( p^{311} T^{10} + \)\(31\!\cdots\!68\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 7 | \( 1 + \)\(93\!\cdots\!44\)\( p^{2} T + \)\(18\!\cdots\!74\)\( p^{4} T^{2} + \)\(66\!\cdots\!52\)\( p^{10} T^{3} + \)\(28\!\cdots\!11\)\( p^{16} T^{4} + \)\(18\!\cdots\!88\)\( p^{24} T^{5} + \)\(11\!\cdots\!88\)\( p^{32} T^{6} + \)\(18\!\cdots\!88\)\( p^{101} T^{7} + \)\(28\!\cdots\!11\)\( p^{170} T^{8} + \)\(66\!\cdots\!52\)\( p^{241} T^{9} + \)\(18\!\cdots\!74\)\( p^{312} T^{10} + \)\(93\!\cdots\!44\)\( p^{387} T^{11} + p^{462} T^{12} \) | |
| 11 | \( 1 + \)\(31\!\cdots\!76\)\( T + \)\(58\!\cdots\!54\)\( p^{2} T^{2} + \)\(15\!\cdots\!12\)\( p^{5} T^{3} + \)\(96\!\cdots\!93\)\( p^{9} T^{4} + \)\(26\!\cdots\!60\)\( p^{11} T^{5} + \)\(10\!\cdots\!28\)\( p^{15} T^{6} + \)\(26\!\cdots\!60\)\( p^{88} T^{7} + \)\(96\!\cdots\!93\)\( p^{163} T^{8} + \)\(15\!\cdots\!12\)\( p^{236} T^{9} + \)\(58\!\cdots\!54\)\( p^{310} T^{10} + \)\(31\!\cdots\!76\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 13 | \( 1 + \)\(82\!\cdots\!04\)\( p T + \)\(13\!\cdots\!70\)\( p^{3} T^{2} + \)\(62\!\cdots\!12\)\( p^{5} T^{3} + \)\(60\!\cdots\!95\)\( p^{7} T^{4} + \)\(17\!\cdots\!28\)\( p^{10} T^{5} + \)\(57\!\cdots\!08\)\( p^{15} T^{6} + \)\(17\!\cdots\!28\)\( p^{87} T^{7} + \)\(60\!\cdots\!95\)\( p^{161} T^{8} + \)\(62\!\cdots\!12\)\( p^{236} T^{9} + \)\(13\!\cdots\!70\)\( p^{311} T^{10} + \)\(82\!\cdots\!04\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 17 | \( 1 - \)\(20\!\cdots\!28\)\( T + \)\(94\!\cdots\!66\)\( p T^{2} - \)\(83\!\cdots\!80\)\( p^{2} T^{3} + \)\(12\!\cdots\!55\)\( p^{4} T^{4} - \)\(24\!\cdots\!16\)\( p^{7} T^{5} + \)\(27\!\cdots\!56\)\( p^{10} T^{6} - \)\(24\!\cdots\!16\)\( p^{84} T^{7} + \)\(12\!\cdots\!55\)\( p^{158} T^{8} - \)\(83\!\cdots\!80\)\( p^{233} T^{9} + \)\(94\!\cdots\!66\)\( p^{309} T^{10} - \)\(20\!\cdots\!28\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 19 | \( 1 + \)\(97\!\cdots\!92\)\( p T + \)\(17\!\cdots\!38\)\( p^{3} T^{2} + \)\(53\!\cdots\!56\)\( p^{6} T^{3} + \)\(81\!\cdots\!77\)\( p^{7} T^{4} + \)\(41\!\cdots\!68\)\( p^{9} T^{5} + \)\(23\!\cdots\!32\)\( p^{11} T^{6} + \)\(41\!\cdots\!68\)\( p^{86} T^{7} + \)\(81\!\cdots\!77\)\( p^{161} T^{8} + \)\(53\!\cdots\!56\)\( p^{237} T^{9} + \)\(17\!\cdots\!38\)\( p^{311} T^{10} + \)\(97\!\cdots\!92\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 23 | \( 1 + \)\(10\!\cdots\!48\)\( T + \)\(10\!\cdots\!02\)\( p T^{2} + \)\(47\!\cdots\!84\)\( p^{2} T^{3} + \)\(11\!\cdots\!91\)\( p^{4} T^{4} + \)\(20\!\cdots\!76\)\( p^{6} T^{5} + \)\(15\!\cdots\!24\)\( p^{9} T^{6} + \)\(20\!\cdots\!76\)\( p^{83} T^{7} + \)\(11\!\cdots\!91\)\( p^{158} T^{8} + \)\(47\!\cdots\!84\)\( p^{233} T^{9} + \)\(10\!\cdots\!02\)\( p^{309} T^{10} + \)\(10\!\cdots\!48\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 29 | \( 1 + \)\(35\!\cdots\!36\)\( T + \)\(18\!\cdots\!66\)\( T^{2} + \)\(11\!\cdots\!92\)\( p T^{3} + \)\(11\!\cdots\!87\)\( p^{2} T^{4} + \)\(29\!\cdots\!04\)\( p^{3} T^{5} + \)\(14\!\cdots\!28\)\( p^{5} T^{6} + \)\(29\!\cdots\!04\)\( p^{80} T^{7} + \)\(11\!\cdots\!87\)\( p^{156} T^{8} + \)\(11\!\cdots\!92\)\( p^{232} T^{9} + \)\(18\!\cdots\!66\)\( p^{308} T^{10} + \)\(35\!\cdots\!36\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 31 | \( 1 + \)\(44\!\cdots\!04\)\( T + \)\(90\!\cdots\!98\)\( p T^{2} + \)\(79\!\cdots\!32\)\( p^{2} T^{3} + \)\(33\!\cdots\!23\)\( p^{4} T^{4} + \)\(75\!\cdots\!16\)\( p^{6} T^{5} + \)\(28\!\cdots\!12\)\( p^{8} T^{6} + \)\(75\!\cdots\!16\)\( p^{83} T^{7} + \)\(33\!\cdots\!23\)\( p^{158} T^{8} + \)\(79\!\cdots\!32\)\( p^{233} T^{9} + \)\(90\!\cdots\!98\)\( p^{309} T^{10} + \)\(44\!\cdots\!04\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 37 | \( 1 - \)\(20\!\cdots\!72\)\( p T + \)\(28\!\cdots\!46\)\( p^{2} T^{2} - \)\(24\!\cdots\!84\)\( p^{3} T^{3} + \)\(49\!\cdots\!63\)\( p^{5} T^{4} - \)\(73\!\cdots\!84\)\( p^{7} T^{5} + \)\(12\!\cdots\!24\)\( p^{9} T^{6} - \)\(73\!\cdots\!84\)\( p^{84} T^{7} + \)\(49\!\cdots\!63\)\( p^{159} T^{8} - \)\(24\!\cdots\!84\)\( p^{234} T^{9} + \)\(28\!\cdots\!46\)\( p^{310} T^{10} - \)\(20\!\cdots\!72\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 41 | \( 1 - \)\(20\!\cdots\!96\)\( T + \)\(61\!\cdots\!58\)\( T^{2} - \)\(87\!\cdots\!48\)\( T^{3} + \)\(18\!\cdots\!43\)\( T^{4} - \)\(53\!\cdots\!24\)\( p T^{5} + \)\(20\!\cdots\!12\)\( p^{2} T^{6} - \)\(53\!\cdots\!24\)\( p^{78} T^{7} + \)\(18\!\cdots\!43\)\( p^{154} T^{8} - \)\(87\!\cdots\!48\)\( p^{231} T^{9} + \)\(61\!\cdots\!58\)\( p^{308} T^{10} - \)\(20\!\cdots\!96\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 43 | \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(32\!\cdots\!90\)\( T^{2} - \)\(57\!\cdots\!00\)\( p T^{3} + \)\(23\!\cdots\!03\)\( p^{2} T^{4} - \)\(33\!\cdots\!00\)\( p^{3} T^{5} + \)\(99\!\cdots\!20\)\( p^{4} T^{6} - \)\(33\!\cdots\!00\)\( p^{80} T^{7} + \)\(23\!\cdots\!03\)\( p^{156} T^{8} - \)\(57\!\cdots\!00\)\( p^{232} T^{9} + \)\(32\!\cdots\!90\)\( p^{308} T^{10} - \)\(10\!\cdots\!00\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 47 | \( 1 - \)\(33\!\cdots\!20\)\( T + \)\(22\!\cdots\!30\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!81\)\( p T^{4} - \)\(14\!\cdots\!20\)\( p^{3} T^{5} + \)\(16\!\cdots\!80\)\( p^{3} T^{6} - \)\(14\!\cdots\!20\)\( p^{80} T^{7} + \)\(52\!\cdots\!81\)\( p^{155} T^{8} - \)\(12\!\cdots\!40\)\( p^{231} T^{9} + \)\(22\!\cdots\!30\)\( p^{308} T^{10} - \)\(33\!\cdots\!20\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 53 | \( 1 + \)\(48\!\cdots\!48\)\( T + \)\(38\!\cdots\!26\)\( T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!87\)\( p T^{4} + \)\(55\!\cdots\!56\)\( p^{2} T^{5} + \)\(31\!\cdots\!96\)\( p^{3} T^{6} + \)\(55\!\cdots\!56\)\( p^{79} T^{7} + \)\(11\!\cdots\!87\)\( p^{155} T^{8} + \)\(13\!\cdots\!76\)\( p^{231} T^{9} + \)\(38\!\cdots\!26\)\( p^{308} T^{10} + \)\(48\!\cdots\!48\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 59 | \( 1 + \)\(39\!\cdots\!12\)\( T + \)\(10\!\cdots\!62\)\( T^{2} + \)\(61\!\cdots\!56\)\( p T^{3} + \)\(14\!\cdots\!23\)\( p^{2} T^{4} + \)\(69\!\cdots\!92\)\( p^{3} T^{5} + \)\(11\!\cdots\!68\)\( p^{4} T^{6} + \)\(69\!\cdots\!92\)\( p^{80} T^{7} + \)\(14\!\cdots\!23\)\( p^{156} T^{8} + \)\(61\!\cdots\!56\)\( p^{232} T^{9} + \)\(10\!\cdots\!62\)\( p^{308} T^{10} + \)\(39\!\cdots\!12\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 61 | \( 1 - \)\(34\!\cdots\!60\)\( T + \)\(17\!\cdots\!34\)\( p T^{2} + \)\(69\!\cdots\!80\)\( p^{2} T^{3} + \)\(19\!\cdots\!15\)\( p^{3} T^{4} + \)\(17\!\cdots\!80\)\( p^{4} T^{5} + \)\(15\!\cdots\!80\)\( p^{5} T^{6} + \)\(17\!\cdots\!80\)\( p^{81} T^{7} + \)\(19\!\cdots\!15\)\( p^{157} T^{8} + \)\(69\!\cdots\!80\)\( p^{233} T^{9} + \)\(17\!\cdots\!34\)\( p^{309} T^{10} - \)\(34\!\cdots\!60\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 67 | \( 1 - \)\(82\!\cdots\!84\)\( p T + \)\(78\!\cdots\!98\)\( p^{2} T^{2} - \)\(40\!\cdots\!40\)\( p^{3} T^{3} + \)\(21\!\cdots\!55\)\( p^{4} T^{4} - \)\(75\!\cdots\!24\)\( p^{5} T^{5} + \)\(26\!\cdots\!76\)\( p^{6} T^{6} - \)\(75\!\cdots\!24\)\( p^{82} T^{7} + \)\(21\!\cdots\!55\)\( p^{158} T^{8} - \)\(40\!\cdots\!40\)\( p^{234} T^{9} + \)\(78\!\cdots\!98\)\( p^{310} T^{10} - \)\(82\!\cdots\!84\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 71 | \( 1 - \)\(17\!\cdots\!72\)\( T + \)\(13\!\cdots\!86\)\( p T^{2} - \)\(31\!\cdots\!20\)\( p^{2} T^{3} + \)\(16\!\cdots\!45\)\( p^{3} T^{4} - \)\(31\!\cdots\!32\)\( p^{4} T^{5} + \)\(12\!\cdots\!44\)\( p^{5} T^{6} - \)\(31\!\cdots\!32\)\( p^{81} T^{7} + \)\(16\!\cdots\!45\)\( p^{157} T^{8} - \)\(31\!\cdots\!20\)\( p^{233} T^{9} + \)\(13\!\cdots\!86\)\( p^{309} T^{10} - \)\(17\!\cdots\!72\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 73 | \( 1 + \)\(18\!\cdots\!16\)\( p T + \)\(18\!\cdots\!34\)\( p^{2} T^{2} + \)\(46\!\cdots\!68\)\( p^{3} T^{3} + \)\(18\!\cdots\!71\)\( p^{4} T^{4} + \)\(41\!\cdots\!08\)\( p^{5} T^{5} + \)\(13\!\cdots\!48\)\( p^{6} T^{6} + \)\(41\!\cdots\!08\)\( p^{82} T^{7} + \)\(18\!\cdots\!71\)\( p^{158} T^{8} + \)\(46\!\cdots\!68\)\( p^{234} T^{9} + \)\(18\!\cdots\!34\)\( p^{310} T^{10} + \)\(18\!\cdots\!16\)\( p^{386} T^{11} + p^{462} T^{12} \) | |
| 79 | \( 1 - \)\(18\!\cdots\!40\)\( T + \)\(52\!\cdots\!54\)\( T^{2} - \)\(76\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!15\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{5} + \)\(23\!\cdots\!80\)\( T^{6} - \)\(16\!\cdots\!00\)\( p^{77} T^{7} + \)\(14\!\cdots\!15\)\( p^{154} T^{8} - \)\(76\!\cdots\!00\)\( p^{231} T^{9} + \)\(52\!\cdots\!54\)\( p^{308} T^{10} - \)\(18\!\cdots\!40\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 83 | \( 1 - \)\(14\!\cdots\!76\)\( T + \)\(28\!\cdots\!26\)\( T^{2} - \)\(26\!\cdots\!92\)\( T^{3} + \)\(30\!\cdots\!47\)\( T^{4} - \)\(21\!\cdots\!52\)\( T^{5} + \)\(19\!\cdots\!56\)\( T^{6} - \)\(21\!\cdots\!52\)\( p^{77} T^{7} + \)\(30\!\cdots\!47\)\( p^{154} T^{8} - \)\(26\!\cdots\!92\)\( p^{231} T^{9} + \)\(28\!\cdots\!26\)\( p^{308} T^{10} - \)\(14\!\cdots\!76\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 89 | \( 1 + \)\(20\!\cdots\!08\)\( T + \)\(19\!\cdots\!02\)\( T^{2} + \)\(18\!\cdots\!96\)\( T^{3} + \)\(26\!\cdots\!03\)\( T^{4} + \)\(10\!\cdots\!12\)\( T^{5} - \)\(50\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!12\)\( p^{77} T^{7} + \)\(26\!\cdots\!03\)\( p^{154} T^{8} + \)\(18\!\cdots\!96\)\( p^{231} T^{9} + \)\(19\!\cdots\!02\)\( p^{308} T^{10} + \)\(20\!\cdots\!08\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
| 97 | \( 1 - \)\(37\!\cdots\!48\)\( T + \)\(29\!\cdots\!82\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(61\!\cdots\!68\)\( T^{5} + \)\(49\!\cdots\!04\)\( T^{6} - \)\(61\!\cdots\!68\)\( p^{77} T^{7} + \)\(39\!\cdots\!55\)\( p^{154} T^{8} - \)\(46\!\cdots\!20\)\( p^{231} T^{9} + \)\(29\!\cdots\!82\)\( p^{308} T^{10} - \)\(37\!\cdots\!48\)\( p^{385} T^{11} + p^{462} T^{12} \) | |
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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
−6.10313657007912254118380846791, −6.07540670594635067697308628127, −5.63443515782769628605120443273, −5.30923287310861798209498802924, −5.29005340151467174660379945552, −5.13742939557240722050642279748, −4.97642923283664814119936247884, −4.36590628073423308143834251081, −4.27996386807407842558877660992, −4.20054893836164945352136276998, −4.05310478776331215535569442846, −3.80751256816614212025898094875, −3.52257416074084319373365059287, −3.44676551451558636504544544213, −2.99377838655968696964830668940, −2.62830434310532667070780898394, −2.30383876350560900182844891165, −2.28246077465997698999768102766, −1.87456834883767948324844855942, −1.69788206073977359454304504919, −1.30233283531644697165825542269, −1.09763803922200594290575467594, −0.868660676754476816002590600525, −0.77965626835235819509176898307, −0.63033634932089762660259264671, 0, 0, 0, 0, 0, 0, 0.63033634932089762660259264671, 0.77965626835235819509176898307, 0.868660676754476816002590600525, 1.09763803922200594290575467594, 1.30233283531644697165825542269, 1.69788206073977359454304504919, 1.87456834883767948324844855942, 2.28246077465997698999768102766, 2.30383876350560900182844891165, 2.62830434310532667070780898394, 2.99377838655968696964830668940, 3.44676551451558636504544544213, 3.52257416074084319373365059287, 3.80751256816614212025898094875, 4.05310478776331215535569442846, 4.20054893836164945352136276998, 4.27996386807407842558877660992, 4.36590628073423308143834251081, 4.97642923283664814119936247884, 5.13742939557240722050642279748, 5.29005340151467174660379945552, 5.30923287310861798209498802924, 5.63443515782769628605120443273, 6.07540670594635067697308628127, 6.10313657007912254118380846791
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.