Dirichlet series
| L(s) = 1 | + 2.40e7·2-s − 4.53e15·4-s + 1.23e16·5-s − 3.24e21·7-s − 8.44e22·8-s + 2.96e23·10-s + 1.82e26·11-s − 1.38e28·13-s − 7.79e28·14-s + 9.31e30·16-s + 3.05e31·17-s − 9.45e32·19-s − 5.58e31·20-s + 4.38e33·22-s − 5.28e34·23-s − 5.70e35·25-s − 3.31e35·26-s + 1.47e37·28-s + 1.57e37·29-s − 1.06e37·31-s − 5.20e37·32-s + 7.34e38·34-s − 3.99e37·35-s + 1.75e39·37-s − 2.27e40·38-s − 1.03e39·40-s − 2.00e41·41-s + ⋯ |
| L(s) = 1 | + 0.506·2-s − 2.01·4-s + 0.0184·5-s − 0.913·7-s − 0.790·8-s + 0.00936·10-s + 0.506·11-s − 0.542·13-s − 0.462·14-s + 1.83·16-s + 1.28·17-s − 2.33·19-s − 0.0372·20-s + 0.256·22-s − 0.998·23-s − 1.28·25-s − 0.274·26-s + 1.84·28-s + 0.804·29-s − 0.0990·31-s − 0.216·32-s + 0.650·34-s − 0.0168·35-s + 0.180·37-s − 1.18·38-s − 0.0145·40-s − 1.49·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(52-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+51/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(6561\) = \(3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.83142\times 10^{8}\) |
| Root analytic conductor: | \(12.1761\) |
| Motivic weight: | \(51\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 6561,\ (\ :51/2, 51/2, 51/2, 51/2),\ 1)\) |
Particular Values
| \(L(26)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{53}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 3 | \( 1 \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 12021633 p T + 79924394556095 p^{6} T^{2} - 4506076641541315347 p^{15} T^{3} + \)\(28\!\cdots\!43\)\( p^{29} T^{4} - 4506076641541315347 p^{66} T^{5} + 79924394556095 p^{108} T^{6} - 12021633 p^{154} T^{7} + p^{204} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 2462800596677064 p T + \)\(18\!\cdots\!28\)\( p^{5} T^{2} - \)\(26\!\cdots\!52\)\( p^{10} T^{3} + \)\(15\!\cdots\!26\)\( p^{16} T^{4} - \)\(26\!\cdots\!52\)\( p^{61} T^{5} + \)\(18\!\cdots\!28\)\( p^{107} T^{6} - 2462800596677064 p^{154} T^{7} + p^{204} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + \)\(46\!\cdots\!56\)\( p T + \)\(33\!\cdots\!36\)\( p^{3} T^{2} - \)\(19\!\cdots\!04\)\( p^{7} T^{3} - \)\(62\!\cdots\!90\)\( p^{11} T^{4} - \)\(19\!\cdots\!04\)\( p^{58} T^{5} + \)\(33\!\cdots\!36\)\( p^{105} T^{6} + \)\(46\!\cdots\!56\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - \)\(18\!\cdots\!24\)\( T + \)\(33\!\cdots\!72\)\( p T^{2} - \)\(47\!\cdots\!40\)\( p^{3} T^{3} + \)\(31\!\cdots\!66\)\( p^{7} T^{4} - \)\(47\!\cdots\!40\)\( p^{54} T^{5} + \)\(33\!\cdots\!72\)\( p^{103} T^{6} - \)\(18\!\cdots\!24\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!32\)\( p T + \)\(13\!\cdots\!84\)\( p^{2} T^{2} + \)\(90\!\cdots\!84\)\( p^{3} T^{3} + \)\(42\!\cdots\!70\)\( p^{6} T^{4} + \)\(90\!\cdots\!84\)\( p^{54} T^{5} + \)\(13\!\cdots\!84\)\( p^{104} T^{6} + \)\(10\!\cdots\!32\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - \)\(17\!\cdots\!92\)\( p T + \)\(68\!\cdots\!12\)\( p^{2} T^{2} - \)\(83\!\cdots\!40\)\( p^{3} T^{3} + \)\(18\!\cdots\!26\)\( p^{4} T^{4} - \)\(83\!\cdots\!40\)\( p^{54} T^{5} + \)\(68\!\cdots\!12\)\( p^{104} T^{6} - \)\(17\!\cdots\!92\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + \)\(94\!\cdots\!48\)\( T + \)\(48\!\cdots\!48\)\( p T^{2} + \)\(13\!\cdots\!16\)\( p^{2} T^{3} + \)\(10\!\cdots\!66\)\( p^{5} T^{4} + \)\(13\!\cdots\!16\)\( p^{53} T^{5} + \)\(48\!\cdots\!48\)\( p^{103} T^{6} + \)\(94\!\cdots\!48\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + \)\(52\!\cdots\!16\)\( T + \)\(19\!\cdots\!24\)\( p T^{2} + \)\(16\!\cdots\!08\)\( p^{3} T^{3} + \)\(30\!\cdots\!50\)\( p^{5} T^{4} + \)\(16\!\cdots\!08\)\( p^{54} T^{5} + \)\(19\!\cdots\!24\)\( p^{103} T^{6} + \)\(52\!\cdots\!16\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - \)\(15\!\cdots\!36\)\( T + \)\(16\!\cdots\!60\)\( p T^{2} + \)\(21\!\cdots\!48\)\( p^{2} T^{3} + \)\(35\!\cdots\!42\)\( p^{3} T^{4} + \)\(21\!\cdots\!48\)\( p^{53} T^{5} + \)\(16\!\cdots\!60\)\( p^{103} T^{6} - \)\(15\!\cdots\!36\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!16\)\( T + \)\(43\!\cdots\!88\)\( T^{2} - \)\(14\!\cdots\!52\)\( p T^{3} + \)\(16\!\cdots\!94\)\( p^{2} T^{4} - \)\(14\!\cdots\!52\)\( p^{52} T^{5} + \)\(43\!\cdots\!88\)\( p^{102} T^{6} + \)\(10\!\cdots\!16\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - \)\(17\!\cdots\!68\)\( T + \)\(29\!\cdots\!04\)\( p T^{2} - \)\(69\!\cdots\!48\)\( p^{2} T^{3} + \)\(73\!\cdots\!70\)\( p^{3} T^{4} - \)\(69\!\cdots\!48\)\( p^{53} T^{5} + \)\(29\!\cdots\!04\)\( p^{103} T^{6} - \)\(17\!\cdots\!68\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + \)\(48\!\cdots\!84\)\( p T + \)\(18\!\cdots\!28\)\( p^{2} T^{2} + \)\(45\!\cdots\!12\)\( p^{3} T^{3} + \)\(13\!\cdots\!74\)\( p^{4} T^{4} + \)\(45\!\cdots\!12\)\( p^{54} T^{5} + \)\(18\!\cdots\!28\)\( p^{104} T^{6} + \)\(48\!\cdots\!84\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + \)\(44\!\cdots\!20\)\( T + \)\(80\!\cdots\!20\)\( p T^{2} + \)\(98\!\cdots\!60\)\( p^{2} T^{3} + \)\(54\!\cdots\!14\)\( p^{3} T^{4} + \)\(98\!\cdots\!60\)\( p^{53} T^{5} + \)\(80\!\cdots\!20\)\( p^{103} T^{6} + \)\(44\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + \)\(15\!\cdots\!20\)\( p T + \)\(29\!\cdots\!20\)\( p^{2} T^{2} + \)\(31\!\cdots\!40\)\( p^{3} T^{3} + \)\(38\!\cdots\!78\)\( p^{4} T^{4} + \)\(31\!\cdots\!40\)\( p^{54} T^{5} + \)\(29\!\cdots\!20\)\( p^{104} T^{6} + \)\(15\!\cdots\!20\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + \)\(20\!\cdots\!56\)\( T + \)\(39\!\cdots\!72\)\( T^{2} + \)\(46\!\cdots\!96\)\( T^{3} + \)\(52\!\cdots\!50\)\( T^{4} + \)\(46\!\cdots\!96\)\( p^{51} T^{5} + \)\(39\!\cdots\!72\)\( p^{102} T^{6} + \)\(20\!\cdots\!56\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + \)\(49\!\cdots\!68\)\( T + \)\(16\!\cdots\!52\)\( T^{2} + \)\(34\!\cdots\!76\)\( T^{3} + \)\(57\!\cdots\!74\)\( T^{4} + \)\(34\!\cdots\!76\)\( p^{51} T^{5} + \)\(16\!\cdots\!52\)\( p^{102} T^{6} + \)\(49\!\cdots\!68\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(37\!\cdots\!76\)\( T^{2} - \)\(48\!\cdots\!20\)\( T^{3} + \)\(58\!\cdots\!86\)\( T^{4} - \)\(48\!\cdots\!20\)\( p^{51} T^{5} + \)\(37\!\cdots\!76\)\( p^{102} T^{6} - \)\(18\!\cdots\!80\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + \)\(65\!\cdots\!44\)\( T + \)\(37\!\cdots\!08\)\( T^{2} + \)\(77\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!66\)\( T^{4} + \)\(77\!\cdots\!80\)\( p^{51} T^{5} + \)\(37\!\cdots\!08\)\( p^{102} T^{6} + \)\(65\!\cdots\!44\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - \)\(68\!\cdots\!52\)\( T + \)\(27\!\cdots\!48\)\( T^{2} - \)\(71\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} - \)\(71\!\cdots\!64\)\( p^{51} T^{5} + \)\(27\!\cdots\!48\)\( p^{102} T^{6} - \)\(68\!\cdots\!52\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!84\)\( T + \)\(74\!\cdots\!92\)\( T^{2} + \)\(35\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} + \)\(35\!\cdots\!44\)\( p^{51} T^{5} + \)\(74\!\cdots\!92\)\( p^{102} T^{6} + \)\(10\!\cdots\!84\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + \)\(23\!\cdots\!60\)\( T + \)\(41\!\cdots\!16\)\( T^{2} + \)\(36\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!46\)\( T^{4} + \)\(36\!\cdots\!20\)\( p^{51} T^{5} + \)\(41\!\cdots\!16\)\( p^{102} T^{6} + \)\(23\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + \)\(13\!\cdots\!88\)\( T + \)\(24\!\cdots\!40\)\( T^{2} + \)\(20\!\cdots\!72\)\( T^{3} + \)\(24\!\cdots\!06\)\( T^{4} + \)\(20\!\cdots\!72\)\( p^{51} T^{5} + \)\(24\!\cdots\!40\)\( p^{102} T^{6} + \)\(13\!\cdots\!88\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + \)\(42\!\cdots\!72\)\( T + \)\(49\!\cdots\!92\)\( T^{2} + \)\(39\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!34\)\( T^{4} + \)\(39\!\cdots\!64\)\( p^{51} T^{5} + \)\(49\!\cdots\!92\)\( p^{102} T^{6} + \)\(42\!\cdots\!72\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - \)\(73\!\cdots\!36\)\( T + \)\(69\!\cdots\!48\)\( T^{2} - \)\(32\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!06\)\( T^{4} - \)\(32\!\cdots\!40\)\( p^{51} T^{5} + \)\(69\!\cdots\!48\)\( p^{102} T^{6} - \)\(73\!\cdots\!36\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.431624389985571671861189027047, −7.77848950787932175385670673391, −7.72626890713644133088157138454, −7.69037487915663809408991583973, −6.82949670751460372575854279500, −6.59130923165225472498367295697, −6.29436465076895265331547868147, −6.07562448139315665111404220722, −5.90938912130410479375056912269, −5.22781906072180867657064018984, −5.04342826782754538845767738204, −4.72965407688855849174581497773, −4.69935610300988739693487053381, −4.21845760812357232935409616084, −3.81260859632622512874743211012, −3.72561525192700164070300271054, −3.67144953773743457274871008002, −3.03516947628176435047554874824, −2.64798783741906128693501421735, −2.63891258111294203245393711454, −1.89415527016589157007926089072, −1.62294192325938347242040725311, −1.41670665676039652652984779261, −1.26365110263354235918530797295, −0.57144149502770346933487902845, 0, 0, 0, 0, 0.57144149502770346933487902845, 1.26365110263354235918530797295, 1.41670665676039652652984779261, 1.62294192325938347242040725311, 1.89415527016589157007926089072, 2.63891258111294203245393711454, 2.64798783741906128693501421735, 3.03516947628176435047554874824, 3.67144953773743457274871008002, 3.72561525192700164070300271054, 3.81260859632622512874743211012, 4.21845760812357232935409616084, 4.69935610300988739693487053381, 4.72965407688855849174581497773, 5.04342826782754538845767738204, 5.22781906072180867657064018984, 5.90938912130410479375056912269, 6.07562448139315665111404220722, 6.29436465076895265331547868147, 6.59130923165225472498367295697, 6.82949670751460372575854279500, 7.69037487915663809408991583973, 7.72626890713644133088157138454, 7.77848950787932175385670673391, 8.431624389985571671861189027047