Dirichlet series
| L(s) = 1 | − 5.78e6·2-s − 6.23e13·4-s + 3.11e16·5-s − 3.91e19·7-s − 1.00e20·8-s − 1.80e23·10-s + 1.90e24·11-s + 1.26e26·13-s + 2.26e26·14-s + 2.17e28·16-s − 2.10e29·17-s − 1.05e30·19-s − 1.93e30·20-s − 1.10e31·22-s − 1.37e32·23-s − 4.32e32·25-s − 7.34e32·26-s + 2.44e33·28-s + 2.27e34·29-s + 7.57e34·31-s − 2.50e35·32-s + 1.21e36·34-s − 1.21e36·35-s − 1.12e37·37-s + 6.12e36·38-s − 3.14e36·40-s − 1.30e38·41-s + ⋯ |
| L(s) = 1 | − 0.487·2-s − 0.442·4-s + 1.16·5-s − 0.540·7-s − 0.0604·8-s − 0.569·10-s + 0.640·11-s + 0.843·13-s + 0.263·14-s + 1.09·16-s − 2.55·17-s − 0.942·19-s − 0.517·20-s − 0.312·22-s − 1.37·23-s − 0.608·25-s − 0.411·26-s + 0.239·28-s + 0.978·29-s + 0.680·31-s − 1.06·32-s + 1.24·34-s − 0.631·35-s − 1.58·37-s + 0.459·38-s − 0.0705·40-s − 1.64·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+47/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(6561\) = \(3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.51383\times 10^{8}\) |
| Root analytic conductor: | \(11.2212\) |
| Motivic weight: | \(47\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 6561,\ (\ :47/2, 47/2, 47/2, 47/2),\ 1)\) |
Particular Values
| \(L(24)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{49}{2})\) | 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 | $C_2 \wr S_4$ | \( 1 + 723195 p^{3} T + 93562089245 p^{10} T^{2} + 242210623551345 p^{22} T^{3} - 68020972333510731 p^{37} T^{4} + 242210623551345 p^{69} T^{5} + 93562089245 p^{104} T^{6} + 723195 p^{144} T^{7} + p^{188} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 1244587209690888 p^{2} T + \)\(44\!\cdots\!08\)\( p^{5} T^{2} - \)\(32\!\cdots\!16\)\( p^{9} T^{3} + \)\(33\!\cdots\!66\)\( p^{16} T^{4} - \)\(32\!\cdots\!16\)\( p^{56} T^{5} + \)\(44\!\cdots\!08\)\( p^{99} T^{6} - 1244587209690888 p^{143} T^{7} + p^{188} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 799371810732416800 p^{2} T + \)\(27\!\cdots\!00\)\( p^{4} T^{2} + \)\(23\!\cdots\!00\)\( p^{6} T^{3} + \)\(11\!\cdots\!86\)\( p^{11} T^{4} + \)\(23\!\cdots\!00\)\( p^{53} T^{5} + \)\(27\!\cdots\!00\)\( p^{98} T^{6} + 799371810732416800 p^{143} T^{7} + p^{188} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - \)\(17\!\cdots\!92\)\( p T + \)\(20\!\cdots\!48\)\( p^{3} T^{2} - \)\(21\!\cdots\!24\)\( p^{6} T^{3} + \)\(14\!\cdots\!70\)\( p^{9} T^{4} - \)\(21\!\cdots\!24\)\( p^{53} T^{5} + \)\(20\!\cdots\!48\)\( p^{97} T^{6} - \)\(17\!\cdots\!92\)\( p^{142} T^{7} + p^{188} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(51\!\cdots\!40\)\( p T^{2} - \)\(37\!\cdots\!20\)\( p^{3} T^{3} + \)\(41\!\cdots\!42\)\( p^{6} T^{4} - \)\(37\!\cdots\!20\)\( p^{50} T^{5} + \)\(51\!\cdots\!40\)\( p^{95} T^{6} - \)\(12\!\cdots\!20\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(21\!\cdots\!20\)\( p T^{2} + \)\(82\!\cdots\!80\)\( p^{3} T^{3} + \)\(28\!\cdots\!94\)\( p^{5} T^{4} + \)\(82\!\cdots\!80\)\( p^{50} T^{5} + \)\(21\!\cdots\!20\)\( p^{95} T^{6} + \)\(21\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!40\)\( T + \)\(18\!\cdots\!24\)\( p T^{2} + \)\(46\!\cdots\!20\)\( p^{3} T^{3} + \)\(23\!\cdots\!74\)\( p^{5} T^{4} + \)\(46\!\cdots\!20\)\( p^{50} T^{5} + \)\(18\!\cdots\!24\)\( p^{95} T^{6} + \)\(10\!\cdots\!40\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + \)\(59\!\cdots\!60\)\( p T + \)\(87\!\cdots\!20\)\( p^{2} T^{2} + \)\(34\!\cdots\!80\)\( p^{3} T^{3} + \)\(26\!\cdots\!98\)\( p^{4} T^{4} + \)\(34\!\cdots\!80\)\( p^{50} T^{5} + \)\(87\!\cdots\!20\)\( p^{96} T^{6} + \)\(59\!\cdots\!60\)\( p^{142} T^{7} + p^{188} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - \)\(22\!\cdots\!60\)\( T + \)\(46\!\cdots\!84\)\( p T^{2} - \)\(24\!\cdots\!20\)\( p^{2} T^{3} + \)\(33\!\cdots\!74\)\( p^{3} T^{4} - \)\(24\!\cdots\!20\)\( p^{49} T^{5} + \)\(46\!\cdots\!84\)\( p^{95} T^{6} - \)\(22\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - \)\(75\!\cdots\!48\)\( T + \)\(41\!\cdots\!68\)\( p T^{2} - \)\(42\!\cdots\!36\)\( p^{2} T^{3} + \)\(50\!\cdots\!70\)\( p^{3} T^{4} - \)\(42\!\cdots\!36\)\( p^{49} T^{5} + \)\(41\!\cdots\!68\)\( p^{95} T^{6} - \)\(75\!\cdots\!48\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(41\!\cdots\!60\)\( p T^{2} + \)\(68\!\cdots\!20\)\( p^{2} T^{3} + \)\(17\!\cdots\!26\)\( p^{3} T^{4} + \)\(68\!\cdots\!20\)\( p^{49} T^{5} + \)\(41\!\cdots\!60\)\( p^{95} T^{6} + \)\(11\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + \)\(31\!\cdots\!08\)\( p T + \)\(13\!\cdots\!28\)\( p^{2} T^{2} + \)\(30\!\cdots\!56\)\( p^{3} T^{3} + \)\(77\!\cdots\!70\)\( p^{4} T^{4} + \)\(30\!\cdots\!56\)\( p^{50} T^{5} + \)\(13\!\cdots\!28\)\( p^{96} T^{6} + \)\(31\!\cdots\!08\)\( p^{142} T^{7} + p^{188} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!00\)\( p T + \)\(13\!\cdots\!00\)\( p^{2} T^{2} + \)\(84\!\cdots\!00\)\( p^{3} T^{3} + \)\(61\!\cdots\!98\)\( p^{4} T^{4} + \)\(84\!\cdots\!00\)\( p^{50} T^{5} + \)\(13\!\cdots\!00\)\( p^{96} T^{6} + \)\(10\!\cdots\!00\)\( p^{142} T^{7} + p^{188} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + \)\(20\!\cdots\!20\)\( T + \)\(10\!\cdots\!60\)\( T^{2} + \)\(18\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!60\)\( p^{47} T^{5} + \)\(10\!\cdots\!60\)\( p^{94} T^{6} + \)\(20\!\cdots\!20\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + \)\(29\!\cdots\!60\)\( T + \)\(33\!\cdots\!60\)\( T^{2} + \)\(87\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!38\)\( T^{4} + \)\(87\!\cdots\!20\)\( p^{47} T^{5} + \)\(33\!\cdots\!60\)\( p^{94} T^{6} + \)\(29\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + \)\(47\!\cdots\!80\)\( T + \)\(71\!\cdots\!76\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + \)\(24\!\cdots\!60\)\( p^{47} T^{5} + \)\(71\!\cdots\!76\)\( p^{94} T^{6} + \)\(47\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - \)\(62\!\cdots\!88\)\( T + \)\(29\!\cdots\!88\)\( T^{2} - \)\(12\!\cdots\!36\)\( T^{3} + \)\(34\!\cdots\!70\)\( T^{4} - \)\(12\!\cdots\!36\)\( p^{47} T^{5} + \)\(29\!\cdots\!88\)\( p^{94} T^{6} - \)\(62\!\cdots\!88\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(35\!\cdots\!40\)\( T^{2} - \)\(38\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!58\)\( T^{4} - \)\(38\!\cdots\!40\)\( p^{47} T^{5} + \)\(35\!\cdots\!40\)\( p^{94} T^{6} - \)\(18\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + \)\(22\!\cdots\!68\)\( T + \)\(32\!\cdots\!48\)\( T^{2} + \)\(42\!\cdots\!16\)\( T^{3} + \)\(43\!\cdots\!70\)\( T^{4} + \)\(42\!\cdots\!16\)\( p^{47} T^{5} + \)\(32\!\cdots\!48\)\( p^{94} T^{6} + \)\(22\!\cdots\!68\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(14\!\cdots\!80\)\( T^{2} - \)\(90\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!18\)\( T^{4} - \)\(90\!\cdots\!60\)\( p^{47} T^{5} + \)\(14\!\cdots\!80\)\( p^{94} T^{6} - \)\(10\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(11\!\cdots\!36\)\( T^{2} + \)\(68\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!86\)\( T^{4} + \)\(68\!\cdots\!20\)\( p^{47} T^{5} + \)\(11\!\cdots\!36\)\( p^{94} T^{6} + \)\(13\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - \)\(14\!\cdots\!60\)\( T + \)\(39\!\cdots\!40\)\( T^{2} - \)\(29\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!58\)\( T^{4} - \)\(29\!\cdots\!20\)\( p^{47} T^{5} + \)\(39\!\cdots\!40\)\( p^{94} T^{6} - \)\(14\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - \)\(79\!\cdots\!80\)\( T + \)\(72\!\cdots\!16\)\( T^{2} - \)\(58\!\cdots\!60\)\( T^{3} + \)\(76\!\cdots\!46\)\( T^{4} - \)\(58\!\cdots\!60\)\( p^{47} T^{5} + \)\(72\!\cdots\!16\)\( p^{94} T^{6} - \)\(79\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - \)\(95\!\cdots\!20\)\( T + \)\(11\!\cdots\!60\)\( T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!38\)\( T^{4} - \)\(65\!\cdots\!60\)\( p^{47} T^{5} + \)\(11\!\cdots\!60\)\( p^{94} T^{6} - \)\(95\!\cdots\!20\)\( p^{141} T^{7} + p^{188} T^{8} \) | |
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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
−8.681864610790622660855104485618, −8.125916188414435032013473602073, −8.042978497232651608670740526778, −7.976501082254238841568430056123, −6.88953555679555684518578691550, −6.87997770760871855238877251470, −6.47368864537354253092664145843, −6.39444036117361291714390984275, −6.21252169421993395507075390963, −5.76357935859784668155267069387, −5.26260994389718562003197785204, −5.03119849712762214046151690135, −4.86419253534466990988122326116, −4.23607960387633193789318476807, −3.92819677004500914098362123575, −3.89513201925732124911927413341, −3.38684999671004069184653407645, −3.04024770518675104404864015464, −2.82767822461899807369969039060, −2.11034618492272821162541810220, −1.99842236289298093432167142576, −1.92292327156562728193686883674, −1.45214145236988996693042056990, −1.24444822465184357253644947432, −0.860233480098223543338242554875, 0, 0, 0, 0, 0.860233480098223543338242554875, 1.24444822465184357253644947432, 1.45214145236988996693042056990, 1.92292327156562728193686883674, 1.99842236289298093432167142576, 2.11034618492272821162541810220, 2.82767822461899807369969039060, 3.04024770518675104404864015464, 3.38684999671004069184653407645, 3.89513201925732124911927413341, 3.92819677004500914098362123575, 4.23607960387633193789318476807, 4.86419253534466990988122326116, 5.03119849712762214046151690135, 5.26260994389718562003197785204, 5.76357935859784668155267069387, 6.21252169421993395507075390963, 6.39444036117361291714390984275, 6.47368864537354253092664145843, 6.87997770760871855238877251470, 6.88953555679555684518578691550, 7.976501082254238841568430056123, 8.042978497232651608670740526778, 8.125916188414435032013473602073, 8.681864610790622660855104485618