Dirichlet series
| L(s) = 1 | + 1.74e11·2-s + 2.43e19·3-s − 7.64e23·4-s − 9.81e27·5-s + 4.23e30·6-s − 2.99e33·7-s − 1.85e35·8-s + 3.44e38·9-s − 1.70e39·10-s − 2.63e41·11-s − 1.85e43·12-s − 2.61e42·13-s − 5.22e44·14-s − 2.38e47·15-s + 1.02e47·16-s + 5.67e48·17-s + 6.00e49·18-s + 2.17e50·19-s + 7.50e51·20-s − 7.29e52·21-s − 4.58e52·22-s + 6.73e53·23-s − 4.52e54·24-s + 5.76e54·25-s − 4.56e53·26-s + 3.72e57·27-s + 2.29e57·28-s + ⋯ |
| L(s) = 1 | + 0.223·2-s + 3.46·3-s − 1.26·4-s − 2.41·5-s + 0.775·6-s − 1.24·7-s − 0.395·8-s + 7·9-s − 0.540·10-s − 1.92·11-s − 4.38·12-s − 0.0261·13-s − 0.279·14-s − 8.35·15-s + 0.279·16-s + 1.41·17-s + 1.56·18-s + 0.671·19-s + 3.05·20-s − 4.31·21-s − 0.431·22-s + 1.09·23-s − 1.37·24-s + 0.348·25-s − 0.00585·26-s + 10.7·27-s + 1.57·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(80-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+79/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(729\) = \(3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.77890\times 10^{12}\) |
| Root analytic conductor: | \(10.8890\) |
| Motivic weight: | \(79\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 729,\ (\ :[79/2]^{6}),\ 1)\) |
Particular Values
| \(L(40)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{81}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p^{39} T )^{6} \) |
| good | 2 | \( 1 - 21768808887 p^{3} T + \)\(77\!\cdots\!59\)\( p^{10} T^{2} - \)\(20\!\cdots\!33\)\( p^{22} T^{3} + \)\(17\!\cdots\!63\)\( p^{38} T^{4} + \)\(10\!\cdots\!89\)\( p^{57} T^{5} + \)\(20\!\cdots\!81\)\( p^{80} T^{6} + \)\(10\!\cdots\!89\)\( p^{136} T^{7} + \)\(17\!\cdots\!63\)\( p^{196} T^{8} - \)\(20\!\cdots\!33\)\( p^{259} T^{9} + \)\(77\!\cdots\!59\)\( p^{326} T^{10} - 21768808887 p^{398} T^{11} + p^{474} T^{12} \) |
| 5 | \( 1 + \)\(19\!\cdots\!32\)\( p T + \)\(14\!\cdots\!62\)\( p^{4} T^{2} + \)\(32\!\cdots\!68\)\( p^{9} T^{3} + \)\(49\!\cdots\!19\)\( p^{17} T^{4} + \)\(12\!\cdots\!52\)\( p^{26} T^{5} + \)\(57\!\cdots\!28\)\( p^{36} T^{6} + \)\(12\!\cdots\!52\)\( p^{105} T^{7} + \)\(49\!\cdots\!19\)\( p^{175} T^{8} + \)\(32\!\cdots\!68\)\( p^{246} T^{9} + \)\(14\!\cdots\!62\)\( p^{320} T^{10} + \)\(19\!\cdots\!32\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 7 | \( 1 + \)\(42\!\cdots\!04\)\( p T + \)\(67\!\cdots\!86\)\( p^{4} T^{2} + \)\(53\!\cdots\!72\)\( p^{7} T^{3} + \)\(14\!\cdots\!93\)\( p^{13} T^{4} + \)\(39\!\cdots\!24\)\( p^{20} T^{5} + \)\(30\!\cdots\!96\)\( p^{29} T^{6} + \)\(39\!\cdots\!24\)\( p^{99} T^{7} + \)\(14\!\cdots\!93\)\( p^{171} T^{8} + \)\(53\!\cdots\!72\)\( p^{244} T^{9} + \)\(67\!\cdots\!86\)\( p^{320} T^{10} + \)\(42\!\cdots\!04\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 11 | \( 1 + \)\(26\!\cdots\!56\)\( T + \)\(86\!\cdots\!74\)\( p^{2} T^{2} + \)\(14\!\cdots\!92\)\( p^{4} T^{3} + \)\(21\!\cdots\!43\)\( p^{8} T^{4} + \)\(20\!\cdots\!20\)\( p^{13} T^{5} + \)\(24\!\cdots\!88\)\( p^{16} T^{6} + \)\(20\!\cdots\!20\)\( p^{92} T^{7} + \)\(21\!\cdots\!43\)\( p^{166} T^{8} + \)\(14\!\cdots\!92\)\( p^{241} T^{9} + \)\(86\!\cdots\!74\)\( p^{318} T^{10} + \)\(26\!\cdots\!56\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 13 | \( 1 + \)\(20\!\cdots\!48\)\( p T + \)\(17\!\cdots\!30\)\( p^{3} T^{2} + \)\(24\!\cdots\!72\)\( p^{6} T^{3} + \)\(65\!\cdots\!55\)\( p^{9} T^{4} + \)\(14\!\cdots\!84\)\( p^{12} T^{5} + \)\(11\!\cdots\!44\)\( p^{16} T^{6} + \)\(14\!\cdots\!84\)\( p^{91} T^{7} + \)\(65\!\cdots\!55\)\( p^{167} T^{8} + \)\(24\!\cdots\!72\)\( p^{243} T^{9} + \)\(17\!\cdots\!30\)\( p^{319} T^{10} + \)\(20\!\cdots\!48\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 17 | \( 1 - \)\(56\!\cdots\!04\)\( T + \)\(37\!\cdots\!74\)\( p T^{2} - \)\(10\!\cdots\!60\)\( p^{2} T^{3} + \)\(13\!\cdots\!15\)\( p^{5} T^{4} - \)\(10\!\cdots\!64\)\( p^{8} T^{5} + \)\(60\!\cdots\!96\)\( p^{12} T^{6} - \)\(10\!\cdots\!64\)\( p^{87} T^{7} + \)\(13\!\cdots\!15\)\( p^{163} T^{8} - \)\(10\!\cdots\!60\)\( p^{239} T^{9} + \)\(37\!\cdots\!74\)\( p^{317} T^{10} - \)\(56\!\cdots\!04\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 19 | \( 1 - \)\(11\!\cdots\!52\)\( p T + \)\(88\!\cdots\!42\)\( p^{2} T^{2} - \)\(76\!\cdots\!56\)\( p^{4} T^{3} + \)\(67\!\cdots\!17\)\( p^{7} T^{4} - \)\(26\!\cdots\!72\)\( p^{10} T^{5} + \)\(18\!\cdots\!72\)\( p^{13} T^{6} - \)\(26\!\cdots\!72\)\( p^{89} T^{7} + \)\(67\!\cdots\!17\)\( p^{165} T^{8} - \)\(76\!\cdots\!56\)\( p^{241} T^{9} + \)\(88\!\cdots\!42\)\( p^{318} T^{10} - \)\(11\!\cdots\!52\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 23 | \( 1 - \)\(67\!\cdots\!24\)\( T + \)\(20\!\cdots\!14\)\( T^{2} - \)\(97\!\cdots\!56\)\( p^{3} T^{3} + \)\(66\!\cdots\!11\)\( p^{4} T^{4} - \)\(58\!\cdots\!28\)\( p^{6} T^{5} + \)\(51\!\cdots\!96\)\( p^{9} T^{6} - \)\(58\!\cdots\!28\)\( p^{85} T^{7} + \)\(66\!\cdots\!11\)\( p^{162} T^{8} - \)\(97\!\cdots\!56\)\( p^{240} T^{9} + \)\(20\!\cdots\!14\)\( p^{316} T^{10} - \)\(67\!\cdots\!24\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 29 | \( 1 + \)\(88\!\cdots\!24\)\( T + \)\(56\!\cdots\!34\)\( p T^{2} + \)\(12\!\cdots\!92\)\( p^{2} T^{3} + \)\(45\!\cdots\!63\)\( p^{3} T^{4} + \)\(28\!\cdots\!16\)\( p^{5} T^{5} + \)\(26\!\cdots\!88\)\( p^{7} T^{6} + \)\(28\!\cdots\!16\)\( p^{84} T^{7} + \)\(45\!\cdots\!63\)\( p^{161} T^{8} + \)\(12\!\cdots\!92\)\( p^{239} T^{9} + \)\(56\!\cdots\!34\)\( p^{317} T^{10} + \)\(88\!\cdots\!24\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 31 | \( 1 + \)\(28\!\cdots\!24\)\( T + \)\(16\!\cdots\!78\)\( p T^{2} + \)\(68\!\cdots\!12\)\( p^{2} T^{3} + \)\(24\!\cdots\!73\)\( p^{3} T^{4} + \)\(24\!\cdots\!96\)\( p^{5} T^{5} + \)\(22\!\cdots\!32\)\( p^{7} T^{6} + \)\(24\!\cdots\!96\)\( p^{84} T^{7} + \)\(24\!\cdots\!73\)\( p^{161} T^{8} + \)\(68\!\cdots\!12\)\( p^{239} T^{9} + \)\(16\!\cdots\!78\)\( p^{317} T^{10} + \)\(28\!\cdots\!24\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 37 | \( 1 + \)\(17\!\cdots\!64\)\( p T + \)\(38\!\cdots\!02\)\( p^{3} T^{2} + \)\(23\!\cdots\!52\)\( p^{3} T^{3} + \)\(33\!\cdots\!43\)\( p^{5} T^{4} + \)\(13\!\cdots\!88\)\( p^{7} T^{5} + \)\(16\!\cdots\!16\)\( p^{9} T^{6} + \)\(13\!\cdots\!88\)\( p^{86} T^{7} + \)\(33\!\cdots\!43\)\( p^{163} T^{8} + \)\(23\!\cdots\!52\)\( p^{240} T^{9} + \)\(38\!\cdots\!02\)\( p^{319} T^{10} + \)\(17\!\cdots\!64\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 41 | \( 1 + \)\(81\!\cdots\!84\)\( T + \)\(12\!\cdots\!78\)\( T^{2} + \)\(71\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!63\)\( p T^{4} + \)\(16\!\cdots\!56\)\( p^{2} T^{5} + \)\(29\!\cdots\!72\)\( p^{3} T^{6} + \)\(16\!\cdots\!56\)\( p^{81} T^{7} + \)\(16\!\cdots\!63\)\( p^{159} T^{8} + \)\(71\!\cdots\!32\)\( p^{237} T^{9} + \)\(12\!\cdots\!78\)\( p^{316} T^{10} + \)\(81\!\cdots\!84\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 43 | \( 1 + \)\(35\!\cdots\!20\)\( T + \)\(36\!\cdots\!50\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!29\)\( p T^{4} + \)\(11\!\cdots\!40\)\( p^{2} T^{5} + \)\(13\!\cdots\!00\)\( p^{3} T^{6} + \)\(11\!\cdots\!40\)\( p^{81} T^{7} + \)\(19\!\cdots\!29\)\( p^{159} T^{8} + \)\(11\!\cdots\!40\)\( p^{237} T^{9} + \)\(36\!\cdots\!50\)\( p^{316} T^{10} + \)\(35\!\cdots\!20\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 47 | \( 1 + \)\(45\!\cdots\!40\)\( T + \)\(40\!\cdots\!70\)\( T^{2} + \)\(36\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!61\)\( p T^{4} - \)\(79\!\cdots\!20\)\( p^{2} T^{5} + \)\(84\!\cdots\!20\)\( p^{3} T^{6} - \)\(79\!\cdots\!20\)\( p^{81} T^{7} + \)\(15\!\cdots\!61\)\( p^{159} T^{8} + \)\(36\!\cdots\!20\)\( p^{237} T^{9} + \)\(40\!\cdots\!70\)\( p^{316} T^{10} + \)\(45\!\cdots\!40\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 53 | \( 1 + \)\(64\!\cdots\!36\)\( T + \)\(53\!\cdots\!14\)\( T^{2} + \)\(81\!\cdots\!36\)\( p T^{3} + \)\(59\!\cdots\!19\)\( p^{2} T^{4} + \)\(86\!\cdots\!44\)\( p^{3} T^{5} + \)\(41\!\cdots\!08\)\( p^{4} T^{6} + \)\(86\!\cdots\!44\)\( p^{82} T^{7} + \)\(59\!\cdots\!19\)\( p^{160} T^{8} + \)\(81\!\cdots\!36\)\( p^{238} T^{9} + \)\(53\!\cdots\!14\)\( p^{316} T^{10} + \)\(64\!\cdots\!36\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 59 | \( 1 + \)\(31\!\cdots\!68\)\( T + \)\(23\!\cdots\!42\)\( T^{2} + \)\(20\!\cdots\!64\)\( p T^{3} + \)\(68\!\cdots\!23\)\( p^{2} T^{4} + \)\(98\!\cdots\!88\)\( p^{3} T^{5} + \)\(15\!\cdots\!08\)\( p^{4} T^{6} + \)\(98\!\cdots\!88\)\( p^{82} T^{7} + \)\(68\!\cdots\!23\)\( p^{160} T^{8} + \)\(20\!\cdots\!64\)\( p^{238} T^{9} + \)\(23\!\cdots\!42\)\( p^{316} T^{10} + \)\(31\!\cdots\!68\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 61 | \( 1 + \)\(93\!\cdots\!40\)\( T + \)\(10\!\cdots\!14\)\( p T^{2} + \)\(93\!\cdots\!80\)\( p^{2} T^{3} + \)\(67\!\cdots\!15\)\( p^{3} T^{4} + \)\(43\!\cdots\!80\)\( p^{4} T^{5} + \)\(25\!\cdots\!80\)\( p^{5} T^{6} + \)\(43\!\cdots\!80\)\( p^{83} T^{7} + \)\(67\!\cdots\!15\)\( p^{161} T^{8} + \)\(93\!\cdots\!80\)\( p^{239} T^{9} + \)\(10\!\cdots\!14\)\( p^{317} T^{10} + \)\(93\!\cdots\!40\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 67 | \( 1 - \)\(32\!\cdots\!24\)\( T + \)\(16\!\cdots\!74\)\( p T^{2} - \)\(54\!\cdots\!60\)\( p^{2} T^{3} + \)\(16\!\cdots\!85\)\( p^{3} T^{4} - \)\(39\!\cdots\!44\)\( p^{4} T^{5} + \)\(85\!\cdots\!48\)\( p^{5} T^{6} - \)\(39\!\cdots\!44\)\( p^{83} T^{7} + \)\(16\!\cdots\!85\)\( p^{161} T^{8} - \)\(54\!\cdots\!60\)\( p^{239} T^{9} + \)\(16\!\cdots\!74\)\( p^{317} T^{10} - \)\(32\!\cdots\!24\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 71 | \( 1 + \)\(12\!\cdots\!68\)\( T + \)\(94\!\cdots\!26\)\( p T^{2} + \)\(11\!\cdots\!80\)\( p^{2} T^{3} + \)\(61\!\cdots\!45\)\( p^{3} T^{4} + \)\(62\!\cdots\!68\)\( p^{4} T^{5} + \)\(27\!\cdots\!64\)\( p^{5} T^{6} + \)\(62\!\cdots\!68\)\( p^{83} T^{7} + \)\(61\!\cdots\!45\)\( p^{161} T^{8} + \)\(11\!\cdots\!80\)\( p^{239} T^{9} + \)\(94\!\cdots\!26\)\( p^{317} T^{10} + \)\(12\!\cdots\!68\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 73 | \( 1 + \)\(10\!\cdots\!12\)\( p T + \)\(16\!\cdots\!46\)\( p^{2} T^{2} + \)\(11\!\cdots\!84\)\( p^{3} T^{3} + \)\(10\!\cdots\!51\)\( p^{4} T^{4} + \)\(61\!\cdots\!16\)\( p^{5} T^{5} + \)\(41\!\cdots\!32\)\( p^{6} T^{6} + \)\(61\!\cdots\!16\)\( p^{84} T^{7} + \)\(10\!\cdots\!51\)\( p^{162} T^{8} + \)\(11\!\cdots\!84\)\( p^{240} T^{9} + \)\(16\!\cdots\!46\)\( p^{318} T^{10} + \)\(10\!\cdots\!12\)\( p^{396} T^{11} + p^{474} T^{12} \) | |
| 79 | \( 1 + \)\(62\!\cdots\!20\)\( T + \)\(12\!\cdots\!14\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!15\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(14\!\cdots\!00\)\( p^{79} T^{7} + \)\(25\!\cdots\!15\)\( p^{158} T^{8} + \)\(11\!\cdots\!00\)\( p^{237} T^{9} + \)\(12\!\cdots\!14\)\( p^{316} T^{10} + \)\(62\!\cdots\!20\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 83 | \( 1 + \)\(19\!\cdots\!68\)\( T + \)\(12\!\cdots\!54\)\( T^{2} - \)\(52\!\cdots\!36\)\( T^{3} + \)\(85\!\cdots\!07\)\( T^{4} - \)\(52\!\cdots\!44\)\( T^{5} + \)\(43\!\cdots\!04\)\( T^{6} - \)\(52\!\cdots\!44\)\( p^{79} T^{7} + \)\(85\!\cdots\!07\)\( p^{158} T^{8} - \)\(52\!\cdots\!36\)\( p^{237} T^{9} + \)\(12\!\cdots\!54\)\( p^{316} T^{10} + \)\(19\!\cdots\!68\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 89 | \( 1 - \)\(25\!\cdots\!28\)\( T + \)\(67\!\cdots\!22\)\( T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!03\)\( T^{4} - \)\(19\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!48\)\( T^{6} - \)\(19\!\cdots\!92\)\( p^{79} T^{7} + \)\(16\!\cdots\!03\)\( p^{158} T^{8} - \)\(10\!\cdots\!16\)\( p^{237} T^{9} + \)\(67\!\cdots\!22\)\( p^{316} T^{10} - \)\(25\!\cdots\!28\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
| 97 | \( 1 - \)\(29\!\cdots\!24\)\( T + \)\(36\!\cdots\!38\)\( T^{2} - \)\(94\!\cdots\!40\)\( T^{3} + \)\(63\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!24\)\( T^{5} + \)\(69\!\cdots\!36\)\( T^{6} - \)\(14\!\cdots\!24\)\( p^{79} T^{7} + \)\(63\!\cdots\!55\)\( p^{158} T^{8} - \)\(94\!\cdots\!40\)\( p^{237} T^{9} + \)\(36\!\cdots\!38\)\( p^{316} T^{10} - \)\(29\!\cdots\!24\)\( p^{395} T^{11} + p^{474} T^{12} \) | |
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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
−6.15698680359189266748488080113, −5.67805869334648619283874919606, −5.29738782485372144189053603041, −5.12456500878090096156460371630, −5.01670999283110271037215160116, −4.93880938435994392027893640589, −4.75600623249159502792324505688, −4.04578124457041586311458611954, −3.96818045826533955827234802604, −3.92193897365363589831069720610, −3.72205164463221403136598426969, −3.70816882875040990064858310193, −3.33933869950605907923198784972, −3.30298221015649698860031318827, −3.02420905090079458446254134456, −2.99884170392942373655563666730, −2.79606695122942687311928476447, −2.25696654238298981270011572819, −2.18757472471178018233807500376, −2.07607810423709834561507246776, −1.64017727124425157025854290439, −1.48398477352298294796244165760, −1.32545232750804397919551328720, −1.14289394885119450613240085563, −0.800842337273365127315693745874, 0, 0, 0, 0, 0, 0, 0.800842337273365127315693745874, 1.14289394885119450613240085563, 1.32545232750804397919551328720, 1.48398477352298294796244165760, 1.64017727124425157025854290439, 2.07607810423709834561507246776, 2.18757472471178018233807500376, 2.25696654238298981270011572819, 2.79606695122942687311928476447, 2.99884170392942373655563666730, 3.02420905090079458446254134456, 3.30298221015649698860031318827, 3.33933869950605907923198784972, 3.70816882875040990064858310193, 3.72205164463221403136598426969, 3.92193897365363589831069720610, 3.96818045826533955827234802604, 4.04578124457041586311458611954, 4.75600623249159502792324505688, 4.93880938435994392027893640589, 5.01670999283110271037215160116, 5.12456500878090096156460371630, 5.29738782485372144189053603041, 5.67805869334648619283874919606, 6.15698680359189266748488080113
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.