Dirichlet series
| L(s) = 1 | + 7.91e6·2-s − 2.57e15·4-s + 1.03e18·5-s − 7.25e21·7-s − 1.17e23·8-s + 8.16e24·10-s − 3.20e26·11-s − 3.14e28·13-s − 5.74e28·14-s + 1.12e30·16-s + 4.39e31·17-s − 1.87e32·19-s − 2.66e33·20-s − 2.53e33·22-s + 3.53e33·23-s − 6.94e34·25-s − 2.48e35·26-s + 1.87e37·28-s − 2.36e37·29-s + 1.66e38·31-s + 3.12e38·32-s + 3.47e38·34-s − 7.48e39·35-s − 1.26e40·37-s − 1.48e39·38-s − 1.21e41·40-s − 2.08e41·41-s + ⋯ |
| L(s) = 1 | + 0.166·2-s − 1.14·4-s + 1.54·5-s − 2.04·7-s − 1.10·8-s + 0.258·10-s − 0.891·11-s − 1.23·13-s − 0.341·14-s + 0.221·16-s + 1.84·17-s − 0.463·19-s − 1.77·20-s − 0.148·22-s + 0.0668·23-s − 0.156·25-s − 0.206·26-s + 2.34·28-s − 1.20·29-s + 1.55·31-s + 1.29·32-s + 0.308·34-s − 3.16·35-s − 1.29·37-s − 0.0772·38-s − 1.70·40-s − 1.55·41-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 | $C_2 \wr S_4$ | \( 1 - 494649 p^{4} T + 645159689375 p^{12} T^{2} + 18228781690938639 p^{22} T^{3} + 30250386207592339683 p^{37} T^{4} + 18228781690938639 p^{73} T^{5} + 645159689375 p^{114} T^{6} - 494649 p^{157} T^{7} + p^{204} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 41274865117664136 p^{2} T + \)\(72\!\cdots\!56\)\( p^{6} T^{2} - \)\(79\!\cdots\!04\)\( p^{11} T^{3} + \)\(45\!\cdots\!14\)\( p^{17} T^{4} - \)\(79\!\cdots\!04\)\( p^{62} T^{5} + \)\(72\!\cdots\!56\)\( p^{108} T^{6} - 41274865117664136 p^{155} T^{7} + p^{204} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!24\)\( p T + \)\(13\!\cdots\!56\)\( p^{3} T^{2} + \)\(41\!\cdots\!92\)\( p^{8} T^{3} + \)\(45\!\cdots\!30\)\( p^{11} T^{4} + \)\(41\!\cdots\!92\)\( p^{59} T^{5} + \)\(13\!\cdots\!56\)\( p^{105} T^{6} + \)\(10\!\cdots\!24\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + \)\(29\!\cdots\!16\)\( p T + \)\(23\!\cdots\!32\)\( p^{2} T^{2} + \)\(63\!\cdots\!00\)\( p^{5} T^{3} + \)\(20\!\cdots\!26\)\( p^{8} T^{4} + \)\(63\!\cdots\!00\)\( p^{56} T^{5} + \)\(23\!\cdots\!32\)\( p^{104} T^{6} + \)\(29\!\cdots\!16\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + \)\(24\!\cdots\!88\)\( p T + \)\(78\!\cdots\!24\)\( p^{2} T^{2} + \)\(13\!\cdots\!96\)\( p^{3} T^{3} + \)\(22\!\cdots\!50\)\( p^{6} T^{4} + \)\(13\!\cdots\!96\)\( p^{54} T^{5} + \)\(78\!\cdots\!24\)\( p^{104} T^{6} + \)\(24\!\cdots\!88\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - \)\(25\!\cdots\!08\)\( p T + \)\(35\!\cdots\!12\)\( p^{2} T^{2} - \)\(13\!\cdots\!60\)\( p^{3} T^{3} - \)\(26\!\cdots\!74\)\( p^{4} T^{4} - \)\(13\!\cdots\!60\)\( p^{54} T^{5} + \)\(35\!\cdots\!12\)\( p^{104} T^{6} - \)\(25\!\cdots\!08\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + \)\(18\!\cdots\!48\)\( T + \)\(14\!\cdots\!48\)\( p T^{2} + \)\(39\!\cdots\!16\)\( p^{2} T^{3} + \)\(33\!\cdots\!54\)\( p^{4} T^{4} + \)\(39\!\cdots\!16\)\( p^{53} T^{5} + \)\(14\!\cdots\!48\)\( p^{103} T^{6} + \)\(18\!\cdots\!48\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - \)\(15\!\cdots\!72\)\( p T + \)\(11\!\cdots\!08\)\( p^{2} T^{2} - \)\(13\!\cdots\!68\)\( p^{3} T^{3} + \)\(30\!\cdots\!90\)\( p^{5} T^{4} - \)\(13\!\cdots\!68\)\( p^{54} T^{5} + \)\(11\!\cdots\!08\)\( p^{104} T^{6} - \)\(15\!\cdots\!72\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + \)\(81\!\cdots\!36\)\( p T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(73\!\cdots\!92\)\( p T^{3} + \)\(87\!\cdots\!58\)\( p^{2} T^{4} + \)\(73\!\cdots\!92\)\( p^{52} T^{5} + \)\(13\!\cdots\!20\)\( p^{102} T^{6} + \)\(81\!\cdots\!36\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(43\!\cdots\!28\)\( T^{2} - \)\(54\!\cdots\!52\)\( p^{2} T^{3} + \)\(76\!\cdots\!74\)\( p^{2} T^{4} - \)\(54\!\cdots\!52\)\( p^{53} T^{5} + \)\(43\!\cdots\!28\)\( p^{102} T^{6} - \)\(16\!\cdots\!04\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + \)\(12\!\cdots\!08\)\( T + \)\(89\!\cdots\!84\)\( p T^{2} + \)\(21\!\cdots\!88\)\( p^{2} T^{3} + \)\(90\!\cdots\!90\)\( p^{3} T^{4} + \)\(21\!\cdots\!88\)\( p^{53} T^{5} + \)\(89\!\cdots\!84\)\( p^{103} T^{6} + \)\(12\!\cdots\!08\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + \)\(20\!\cdots\!24\)\( T + \)\(15\!\cdots\!68\)\( p T^{2} + \)\(52\!\cdots\!52\)\( p^{2} T^{3} + \)\(23\!\cdots\!74\)\( p^{3} T^{4} + \)\(52\!\cdots\!52\)\( p^{53} T^{5} + \)\(15\!\cdots\!68\)\( p^{103} T^{6} + \)\(20\!\cdots\!24\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - \)\(17\!\cdots\!60\)\( T + \)\(27\!\cdots\!20\)\( p T^{2} + \)\(26\!\cdots\!20\)\( p^{2} T^{3} - \)\(43\!\cdots\!86\)\( p^{3} T^{4} + \)\(26\!\cdots\!20\)\( p^{53} T^{5} + \)\(27\!\cdots\!20\)\( p^{103} T^{6} - \)\(17\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - \)\(86\!\cdots\!40\)\( T + \)\(19\!\cdots\!40\)\( p T^{2} - \)\(22\!\cdots\!80\)\( p^{2} T^{3} + \)\(25\!\cdots\!66\)\( p^{3} T^{4} - \)\(22\!\cdots\!80\)\( p^{53} T^{5} + \)\(19\!\cdots\!40\)\( p^{103} T^{6} - \)\(86\!\cdots\!40\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - \)\(36\!\cdots\!76\)\( T + \)\(76\!\cdots\!92\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!30\)\( T^{4} - \)\(10\!\cdots\!96\)\( p^{51} T^{5} + \)\(76\!\cdots\!92\)\( p^{102} T^{6} - \)\(36\!\cdots\!76\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - \)\(39\!\cdots\!32\)\( T + \)\(96\!\cdots\!92\)\( T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!94\)\( T^{4} - \)\(17\!\cdots\!64\)\( p^{51} T^{5} + \)\(96\!\cdots\!92\)\( p^{102} T^{6} - \)\(39\!\cdots\!32\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + \)\(20\!\cdots\!60\)\( p T + \)\(94\!\cdots\!56\)\( T^{2} + \)\(79\!\cdots\!40\)\( p T^{3} + \)\(19\!\cdots\!26\)\( T^{4} + \)\(79\!\cdots\!40\)\( p^{52} T^{5} + \)\(94\!\cdots\!56\)\( p^{102} T^{6} + \)\(20\!\cdots\!60\)\( p^{154} T^{7} + p^{204} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - \)\(38\!\cdots\!04\)\( T + \)\(44\!\cdots\!88\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!00\)\( p^{51} T^{5} + \)\(44\!\cdots\!88\)\( p^{102} T^{6} - \)\(38\!\cdots\!04\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - \)\(57\!\cdots\!72\)\( T + \)\(49\!\cdots\!28\)\( T^{2} - \)\(33\!\cdots\!64\)\( T^{3} + \)\(18\!\cdots\!70\)\( T^{4} - \)\(33\!\cdots\!64\)\( p^{51} T^{5} + \)\(49\!\cdots\!28\)\( p^{102} T^{6} - \)\(57\!\cdots\!72\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - \)\(16\!\cdots\!84\)\( T + \)\(11\!\cdots\!72\)\( T^{2} - \)\(25\!\cdots\!04\)\( T^{3} + \)\(76\!\cdots\!30\)\( T^{4} - \)\(25\!\cdots\!04\)\( p^{51} T^{5} + \)\(11\!\cdots\!72\)\( p^{102} T^{6} - \)\(16\!\cdots\!84\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + \)\(74\!\cdots\!00\)\( T + \)\(36\!\cdots\!16\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!46\)\( T^{4} + \)\(11\!\cdots\!00\)\( p^{51} T^{5} + \)\(36\!\cdots\!16\)\( p^{102} T^{6} + \)\(74\!\cdots\!00\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - \)\(18\!\cdots\!88\)\( T + \)\(18\!\cdots\!20\)\( T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(76\!\cdots\!66\)\( T^{4} - \)\(10\!\cdots\!92\)\( p^{51} T^{5} + \)\(18\!\cdots\!20\)\( p^{102} T^{6} - \)\(18\!\cdots\!88\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - \)\(38\!\cdots\!68\)\( T + \)\(10\!\cdots\!72\)\( T^{2} - \)\(29\!\cdots\!96\)\( T^{3} + \)\(38\!\cdots\!74\)\( T^{4} - \)\(29\!\cdots\!96\)\( p^{51} T^{5} + \)\(10\!\cdots\!72\)\( p^{102} T^{6} - \)\(38\!\cdots\!68\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - \)\(61\!\cdots\!44\)\( T + \)\(86\!\cdots\!88\)\( T^{2} - \)\(35\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!26\)\( T^{4} - \)\(35\!\cdots\!20\)\( p^{51} T^{5} + \)\(86\!\cdots\!88\)\( p^{102} T^{6} - \)\(61\!\cdots\!44\)\( p^{153} T^{7} + p^{204} T^{8} \) | |
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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
−8.647571543614444594232070294982, −7.86922788031285324685395783457, −7.53740910889910503276603834540, −7.41622166401894642095986184316, −6.93157319017144343070464026898, −6.69489682665420334372548887263, −6.09383304265595921056008529378, −6.00529105067081039951174622305, −5.94050739413912526539014938036, −5.52817868414719136276613548611, −5.12002699542144466327956335941, −4.99868542348145412926622776415, −4.79676819407361333458190879302, −4.05537419113074308178480106896, −3.85929248881461261858867451732, −3.50869125229758383329814142632, −3.39563104038811018527659988051, −3.04228364797455700154395160291, −2.46886767347096721204867431632, −2.38281851915308035274392635270, −2.33050102151324831582246592962, −1.91990594498892760476820371587, −1.15419031310297506357025280665, −1.03066736881582735587204639871, −0.908489985763106483216255553255, 0, 0, 0, 0, 0.908489985763106483216255553255, 1.03066736881582735587204639871, 1.15419031310297506357025280665, 1.91990594498892760476820371587, 2.33050102151324831582246592962, 2.38281851915308035274392635270, 2.46886767347096721204867431632, 3.04228364797455700154395160291, 3.39563104038811018527659988051, 3.50869125229758383329814142632, 3.85929248881461261858867451732, 4.05537419113074308178480106896, 4.79676819407361333458190879302, 4.99868542348145412926622776415, 5.12002699542144466327956335941, 5.52817868414719136276613548611, 5.94050739413912526539014938036, 6.00529105067081039951174622305, 6.09383304265595921056008529378, 6.69489682665420334372548887263, 6.93157319017144343070464026898, 7.41622166401894642095986184316, 7.53740910889910503276603834540, 7.86922788031285324685395783457, 8.647571543614444594232070294982