Dirichlet series
| L(s) = 1 | + 33·2-s − 53·4-s + 570·5-s − 3.23e3·7-s − 1.77e4·8-s + 1.88e4·10-s − 9.66e4·11-s − 1.41e5·13-s − 1.06e5·14-s − 4.00e5·16-s + 2.85e5·17-s − 4.65e5·19-s − 3.02e4·20-s − 3.19e6·22-s − 2.37e5·23-s − 2.03e6·25-s − 4.65e6·26-s + 1.71e5·28-s + 2.80e6·29-s − 8.24e6·31-s − 2.47e6·32-s + 9.41e6·34-s − 1.84e6·35-s − 8.20e6·37-s − 1.53e7·38-s − 1.01e7·40-s + 2.88e7·41-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.103·4-s + 0.407·5-s − 0.509·7-s − 1.53·8-s + 0.594·10-s − 1.99·11-s − 1.37·13-s − 0.743·14-s − 1.52·16-s + 0.828·17-s − 0.818·19-s − 0.0422·20-s − 2.90·22-s − 0.176·23-s − 1.04·25-s − 1.99·26-s + 0.0527·28-s + 0.736·29-s − 1.60·31-s − 0.417·32-s + 1.20·34-s − 0.207·35-s − 0.719·37-s − 1.19·38-s − 0.625·40-s + 1.59·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(43046721\) = \(3^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.02893\times 10^{6}\) |
| Root analytic conductor: | \(6.45893\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 43046721,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | $C_2 \wr S_4$ | \( 1 - 33 T + 571 p T^{2} - 2709 p^{3} T^{3} + 9225 p^{6} T^{4} - 2709 p^{12} T^{5} + 571 p^{19} T^{6} - 33 p^{27} T^{7} + p^{36} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 114 p T + 2359844 T^{2} - 1041935148 p T^{3} + 92762982621 p^{2} T^{4} - 1041935148 p^{10} T^{5} + 2359844 p^{18} T^{6} - 114 p^{28} T^{7} + p^{36} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 3238 T + 119906236 T^{2} + 58305420946 p T^{3} + 134190217029574 p^{2} T^{4} + 58305420946 p^{10} T^{5} + 119906236 p^{18} T^{6} + 3238 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 8790 p T + 7558569044 T^{2} + 325066480736562 T^{3} + 16839345925797528006 T^{4} + 325066480736562 p^{9} T^{5} + 7558569044 p^{18} T^{6} + 8790 p^{28} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 141118 T + 31246561720 T^{2} + 4533768793498084 T^{3} + \)\(44\!\cdots\!13\)\( T^{4} + 4533768793498084 p^{9} T^{5} + 31246561720 p^{18} T^{6} + 141118 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 285156 T + 226170336074 T^{2} - 67676985549735120 T^{3} + \)\(42\!\cdots\!11\)\( T^{4} - 67676985549735120 p^{9} T^{5} + 226170336074 p^{18} T^{6} - 285156 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 465166 T + 1253122491484 T^{2} + 412658395025562430 T^{3} + \)\(59\!\cdots\!90\)\( T^{4} + 412658395025562430 p^{9} T^{5} + 1253122491484 p^{18} T^{6} + 465166 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 237318 T + 4543399922804 T^{2} + 2498475292049284254 T^{3} + \)\(99\!\cdots\!18\)\( T^{4} + 2498475292049284254 p^{9} T^{5} + 4543399922804 p^{18} T^{6} + 237318 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 2806926 T + 32720851465208 T^{2} - \)\(10\!\cdots\!44\)\( T^{3} + \)\(63\!\cdots\!37\)\( T^{4} - \)\(10\!\cdots\!44\)\( p^{9} T^{5} + 32720851465208 p^{18} T^{6} - 2806926 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 8244448 T + 31193381326492 T^{2} - \)\(17\!\cdots\!84\)\( T^{3} - \)\(15\!\cdots\!82\)\( T^{4} - \)\(17\!\cdots\!84\)\( p^{9} T^{5} + 31193381326492 p^{18} T^{6} + 8244448 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 8204014 T + 515919142675720 T^{2} + \)\(31\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!57\)\( T^{4} + \)\(31\!\cdots\!40\)\( p^{9} T^{5} + 515919142675720 p^{18} T^{6} + 8204014 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 28889292 T + 1138435341360164 T^{2} - \)\(18\!\cdots\!48\)\( T^{3} + \)\(46\!\cdots\!30\)\( T^{4} - \)\(18\!\cdots\!48\)\( p^{9} T^{5} + 1138435341360164 p^{18} T^{6} - 28889292 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 39586246 T + 2466601564125988 T^{2} + \)\(61\!\cdots\!22\)\( T^{3} + \)\(19\!\cdots\!14\)\( T^{4} + \)\(61\!\cdots\!22\)\( p^{9} T^{5} + 2466601564125988 p^{18} T^{6} + 39586246 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 17514588 T + 3035810871500588 T^{2} + \)\(34\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!30\)\( T^{4} + \)\(34\!\cdots\!80\)\( p^{9} T^{5} + 3035810871500588 p^{18} T^{6} + 17514588 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 155826504 T + 13276694126168204 T^{2} + \)\(76\!\cdots\!36\)\( T^{3} + \)\(39\!\cdots\!94\)\( T^{4} + \)\(76\!\cdots\!36\)\( p^{9} T^{5} + 13276694126168204 p^{18} T^{6} + 155826504 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 341256108 T + 68935945770996812 T^{2} - \)\(94\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} - \)\(94\!\cdots\!92\)\( p^{9} T^{5} + 68935945770996812 p^{18} T^{6} - 341256108 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 238298254 T + 51683712635676232 T^{2} + \)\(70\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!17\)\( T^{4} + \)\(70\!\cdots\!20\)\( p^{9} T^{5} + 51683712635676232 p^{18} T^{6} + 238298254 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 72601486 T + 53375901164893468 T^{2} + \)\(33\!\cdots\!10\)\( T^{3} + \)\(20\!\cdots\!98\)\( T^{4} + \)\(33\!\cdots\!10\)\( p^{9} T^{5} + 53375901164893468 p^{18} T^{6} + 72601486 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 667543266 T + 324330981864086300 T^{2} + \)\(10\!\cdots\!38\)\( T^{3} + \)\(25\!\cdots\!90\)\( T^{4} + \)\(10\!\cdots\!38\)\( p^{9} T^{5} + 324330981864086300 p^{18} T^{6} + 667543266 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 520588288 T + 297326185778875726 T^{2} + \)\(91\!\cdots\!48\)\( T^{3} + \)\(28\!\cdots\!15\)\( T^{4} + \)\(91\!\cdots\!48\)\( p^{9} T^{5} + 297326185778875726 p^{18} T^{6} + 520588288 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 137324662 T + 371849866559117524 T^{2} + \)\(43\!\cdots\!50\)\( T^{3} + \)\(63\!\cdots\!14\)\( T^{4} + \)\(43\!\cdots\!50\)\( p^{9} T^{5} + 371849866559117524 p^{18} T^{6} + 137324662 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 278546844 T + 628277248707987644 T^{2} + \)\(10\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!94\)\( T^{4} + \)\(10\!\cdots\!96\)\( p^{9} T^{5} + 628277248707987644 p^{18} T^{6} + 278546844 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 237712932 T + 1048566036298583786 T^{2} - \)\(17\!\cdots\!68\)\( T^{3} + \)\(50\!\cdots\!31\)\( T^{4} - \)\(17\!\cdots\!68\)\( p^{9} T^{5} + 1048566036298583786 p^{18} T^{6} - 237712932 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 700071356 T + 829522214466347476 T^{2} + \)\(12\!\cdots\!40\)\( T^{3} - \)\(78\!\cdots\!98\)\( T^{4} + \)\(12\!\cdots\!40\)\( p^{9} T^{5} + 829522214466347476 p^{18} T^{6} - 700071356 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−9.457633736804716350493634199616, −8.901577577327959308257187808427, −8.707990423411316813092679460510, −8.544926882149707826827137692851, −7.952107288706171012474675416690, −7.72538217072349461557780323218, −7.47025351238493320747530710167, −7.29656574214117293249369867824, −6.66917353055784781458920030061, −6.41281901328202187967985009613, −5.91283152617209240812696729895, −5.83961215569022687987732246723, −5.52704708412951657481920190135, −4.99117634337825505343376026512, −4.77358617986855585045357893347, −4.66680377330581493176074612713, −4.49407857272408160523617612818, −3.62009516182453785484666851439, −3.57858118898967322965834525255, −3.15991720095668085369693804831, −2.80978779201831409454790721014, −2.31648219667725304992026421671, −2.20445491728070375229398836672, −1.47023727254381726705437189765, −1.27689172335539528796919798208, 0, 0, 0, 0, 1.27689172335539528796919798208, 1.47023727254381726705437189765, 2.20445491728070375229398836672, 2.31648219667725304992026421671, 2.80978779201831409454790721014, 3.15991720095668085369693804831, 3.57858118898967322965834525255, 3.62009516182453785484666851439, 4.49407857272408160523617612818, 4.66680377330581493176074612713, 4.77358617986855585045357893347, 4.99117634337825505343376026512, 5.52704708412951657481920190135, 5.83961215569022687987732246723, 5.91283152617209240812696729895, 6.41281901328202187967985009613, 6.66917353055784781458920030061, 7.29656574214117293249369867824, 7.47025351238493320747530710167, 7.72538217072349461557780323218, 7.952107288706171012474675416690, 8.544926882149707826827137692851, 8.707990423411316813092679460510, 8.901577577327959308257187808427, 9.457633736804716350493634199616