Dirichlet series
| L(s) = 1 | − 3.27e7·2-s + 2.19e13·4-s − 1.21e18·5-s + 6.56e21·7-s + 8.19e22·8-s + 3.97e25·10-s − 3.52e26·11-s + 3.07e28·13-s − 2.15e29·14-s − 3.49e30·16-s − 4.81e31·17-s + 8.17e32·19-s − 2.66e31·20-s + 1.15e34·22-s + 5.44e34·23-s − 4.40e35·25-s − 1.00e36·26-s + 1.44e35·28-s − 2.45e37·29-s − 7.40e37·31-s + 1.12e38·32-s + 1.57e39·34-s − 7.97e39·35-s + 9.22e39·37-s − 2.67e40·38-s − 9.95e40·40-s − 1.46e41·41-s + ⋯ |
| L(s) = 1 | − 0.690·2-s + 0.00973·4-s − 1.82·5-s + 1.85·7-s + 0.767·8-s + 1.25·10-s − 0.980·11-s + 1.20·13-s − 1.27·14-s − 0.690·16-s − 2.02·17-s + 2.01·19-s − 0.0177·20-s + 0.676·22-s + 1.02·23-s − 0.991·25-s − 0.833·26-s + 0.0180·28-s − 1.25·29-s − 0.691·31-s + 0.466·32-s + 1.39·34-s − 3.37·35-s + 0.945·37-s − 1.39·38-s − 1.39·40-s − 1.09·41-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 | $C_2 \wr S_4$ | \( 1 + 4094505 p^{3} T + 128299190695 p^{13} T^{2} - 23025772401099165 p^{21} T^{3} - 5761999364104204611 p^{37} T^{4} - 23025772401099165 p^{72} T^{5} + 128299190695 p^{115} T^{6} + 4094505 p^{156} T^{7} + p^{204} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 242822622593511576 p T + \)\(61\!\cdots\!84\)\( p^{5} T^{2} + \)\(14\!\cdots\!44\)\( p^{10} T^{3} + \)\(82\!\cdots\!26\)\( p^{16} T^{4} + \)\(14\!\cdots\!44\)\( p^{61} T^{5} + \)\(61\!\cdots\!84\)\( p^{107} T^{6} + 242822622593511576 p^{154} T^{7} + p^{204} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 - \)\(13\!\cdots\!00\)\( p^{2} T + \)\(48\!\cdots\!00\)\( p^{6} T^{2} - \)\(78\!\cdots\!00\)\( p^{10} T^{3} + \)\(15\!\cdots\!02\)\( p^{14} T^{4} - \)\(78\!\cdots\!00\)\( p^{61} T^{5} + \)\(48\!\cdots\!00\)\( p^{108} T^{6} - \)\(13\!\cdots\!00\)\( p^{155} T^{7} + p^{204} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + \)\(35\!\cdots\!48\)\( T + \)\(25\!\cdots\!48\)\( p^{2} T^{2} + \)\(42\!\cdots\!96\)\( p^{5} T^{3} + \)\(20\!\cdots\!70\)\( p^{9} T^{4} + \)\(42\!\cdots\!96\)\( p^{56} T^{5} + \)\(25\!\cdots\!48\)\( p^{104} T^{6} + \)\(35\!\cdots\!48\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - \)\(18\!\cdots\!20\)\( p^{2} T + \)\(13\!\cdots\!40\)\( p^{2} T^{2} - \)\(25\!\cdots\!80\)\( p^{3} T^{3} + \)\(43\!\cdots\!82\)\( p^{6} T^{4} - \)\(25\!\cdots\!80\)\( p^{54} T^{5} + \)\(13\!\cdots\!40\)\( p^{104} T^{6} - \)\(18\!\cdots\!20\)\( p^{155} T^{7} + p^{204} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + \)\(28\!\cdots\!60\)\( p T + \)\(89\!\cdots\!80\)\( p^{2} T^{2} + \)\(15\!\cdots\!20\)\( p^{3} T^{3} + \)\(27\!\cdots\!18\)\( p^{4} T^{4} + \)\(15\!\cdots\!20\)\( p^{54} T^{5} + \)\(89\!\cdots\!80\)\( p^{104} T^{6} + \)\(28\!\cdots\!60\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - \)\(81\!\cdots\!80\)\( T + \)\(33\!\cdots\!04\)\( p T^{2} - \)\(49\!\cdots\!40\)\( p^{3} T^{3} + \)\(60\!\cdots\!34\)\( p^{5} T^{4} - \)\(49\!\cdots\!40\)\( p^{54} T^{5} + \)\(33\!\cdots\!04\)\( p^{103} T^{6} - \)\(81\!\cdots\!80\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - \)\(23\!\cdots\!60\)\( p T + \)\(17\!\cdots\!60\)\( p^{2} T^{2} - \)\(15\!\cdots\!60\)\( p^{4} T^{3} + \)\(24\!\cdots\!22\)\( p^{6} T^{4} - \)\(15\!\cdots\!60\)\( p^{55} T^{5} + \)\(17\!\cdots\!60\)\( p^{104} T^{6} - \)\(23\!\cdots\!60\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + \)\(24\!\cdots\!20\)\( T + \)\(10\!\cdots\!16\)\( T^{2} + \)\(54\!\cdots\!60\)\( p T^{3} + \)\(59\!\cdots\!06\)\( p^{2} T^{4} + \)\(54\!\cdots\!60\)\( p^{52} T^{5} + \)\(10\!\cdots\!16\)\( p^{102} T^{6} + \)\(24\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + \)\(74\!\cdots\!72\)\( T + \)\(12\!\cdots\!28\)\( p T^{2} + \)\(19\!\cdots\!84\)\( p^{2} T^{3} + \)\(20\!\cdots\!70\)\( p^{3} T^{4} + \)\(19\!\cdots\!84\)\( p^{53} T^{5} + \)\(12\!\cdots\!28\)\( p^{103} T^{6} + \)\(74\!\cdots\!72\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - \)\(92\!\cdots\!60\)\( T + \)\(35\!\cdots\!80\)\( p T^{2} + \)\(42\!\cdots\!80\)\( p^{2} T^{3} - \)\(53\!\cdots\!54\)\( p^{3} T^{4} + \)\(42\!\cdots\!80\)\( p^{53} T^{5} + \)\(35\!\cdots\!80\)\( p^{103} T^{6} - \)\(92\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + \)\(14\!\cdots\!68\)\( T + \)\(71\!\cdots\!28\)\( p T^{2} + \)\(20\!\cdots\!36\)\( p^{2} T^{3} + \)\(10\!\cdots\!70\)\( p^{3} T^{4} + \)\(20\!\cdots\!36\)\( p^{53} T^{5} + \)\(71\!\cdots\!28\)\( p^{103} T^{6} + \)\(14\!\cdots\!68\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + \)\(89\!\cdots\!00\)\( p T + \)\(30\!\cdots\!00\)\( p^{2} T^{2} + \)\(17\!\cdots\!00\)\( p^{3} T^{3} + \)\(40\!\cdots\!98\)\( p^{4} T^{4} + \)\(17\!\cdots\!00\)\( p^{54} T^{5} + \)\(30\!\cdots\!00\)\( p^{104} T^{6} + \)\(89\!\cdots\!00\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + \)\(14\!\cdots\!40\)\( p T + \)\(27\!\cdots\!20\)\( p^{2} T^{2} + \)\(41\!\cdots\!80\)\( p^{3} T^{3} + \)\(33\!\cdots\!78\)\( p^{4} T^{4} + \)\(41\!\cdots\!80\)\( p^{54} T^{5} + \)\(27\!\cdots\!20\)\( p^{104} T^{6} + \)\(14\!\cdots\!40\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - \)\(46\!\cdots\!60\)\( T + \)\(11\!\cdots\!80\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(49\!\cdots\!18\)\( T^{4} + \)\(14\!\cdots\!80\)\( p^{51} T^{5} + \)\(11\!\cdots\!80\)\( p^{102} T^{6} - \)\(46\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!36\)\( T^{2} + \)\(78\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} + \)\(78\!\cdots\!80\)\( p^{51} T^{5} + \)\(43\!\cdots\!36\)\( p^{102} T^{6} + \)\(11\!\cdots\!40\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - \)\(34\!\cdots\!48\)\( T + \)\(36\!\cdots\!08\)\( T^{2} - \)\(87\!\cdots\!96\)\( T^{3} + \)\(55\!\cdots\!70\)\( T^{4} - \)\(87\!\cdots\!96\)\( p^{51} T^{5} + \)\(36\!\cdots\!08\)\( p^{102} T^{6} - \)\(34\!\cdots\!48\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - \)\(30\!\cdots\!20\)\( T + \)\(47\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(91\!\cdots\!78\)\( T^{4} - \)\(12\!\cdots\!60\)\( p^{51} T^{5} + \)\(47\!\cdots\!20\)\( p^{102} T^{6} - \)\(30\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + \)\(39\!\cdots\!88\)\( T + \)\(95\!\cdots\!88\)\( T^{2} + \)\(15\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} + \)\(15\!\cdots\!36\)\( p^{51} T^{5} + \)\(95\!\cdots\!88\)\( p^{102} T^{6} + \)\(39\!\cdots\!88\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(74\!\cdots\!40\)\( T^{2} - \)\(35\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!58\)\( T^{4} - \)\(35\!\cdots\!40\)\( p^{51} T^{5} + \)\(74\!\cdots\!40\)\( p^{102} T^{6} - \)\(10\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - \)\(40\!\cdots\!20\)\( T + \)\(18\!\cdots\!16\)\( T^{2} - \)\(34\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(34\!\cdots\!40\)\( p^{51} T^{5} + \)\(18\!\cdots\!16\)\( p^{102} T^{6} - \)\(40\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!60\)\( T + \)\(32\!\cdots\!20\)\( T^{2} + \)\(23\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!78\)\( T^{4} + \)\(23\!\cdots\!20\)\( p^{51} T^{5} + \)\(32\!\cdots\!20\)\( p^{102} T^{6} + \)\(10\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - \)\(90\!\cdots\!40\)\( T + \)\(11\!\cdots\!56\)\( T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!26\)\( T^{4} - \)\(71\!\cdots\!80\)\( p^{51} T^{5} + \)\(11\!\cdots\!56\)\( p^{102} T^{6} - \)\(90\!\cdots\!40\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(81\!\cdots\!80\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{3} + \)\(68\!\cdots\!18\)\( T^{4} + \)\(24\!\cdots\!60\)\( p^{51} T^{5} + \)\(81\!\cdots\!80\)\( p^{102} T^{6} + \)\(13\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
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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
−8.117235622786015718700923031183, −8.019052315958747913263920483247, −7.81541745757924403724621125094, −7.44434260310396390474560144366, −7.34660204515450063742483334997, −6.77805109331022514764955177228, −6.69746720597513449694905982127, −6.01394777354338659154578819711, −5.83183845521412744351751983144, −5.22932812937185530145181099543, −5.05403491020660092280692353942, −4.79335093923147865247485179781, −4.77138462425557729691056876718, −4.01171020693420282866224155883, −3.91505027022274171287639028902, −3.82046590023320088784991062826, −3.42417053016984942704584908011, −2.95656414471797151613061440652, −2.68915688593315335539210333860, −2.08221250321499118447894972990, −1.96248357791506603776116260041, −1.77304906360554779055802019848, −1.15171686526366272467641712662, −1.08578096437483056115018582810, −1.02957437177077137303450302110, 0, 0, 0, 0, 1.02957437177077137303450302110, 1.08578096437483056115018582810, 1.15171686526366272467641712662, 1.77304906360554779055802019848, 1.96248357791506603776116260041, 2.08221250321499118447894972990, 2.68915688593315335539210333860, 2.95656414471797151613061440652, 3.42417053016984942704584908011, 3.82046590023320088784991062826, 3.91505027022274171287639028902, 4.01171020693420282866224155883, 4.77138462425557729691056876718, 4.79335093923147865247485179781, 5.05403491020660092280692353942, 5.22932812937185530145181099543, 5.83183845521412744351751983144, 6.01394777354338659154578819711, 6.69746720597513449694905982127, 6.77805109331022514764955177228, 7.34660204515450063742483334997, 7.44434260310396390474560144366, 7.81541745757924403724621125094, 8.019052315958747913263920483247, 8.117235622786015718700923031183