Dirichlet series
| L(s) = 1 | − 324·3-s − 240·5-s − 4.84e3·7-s + 6.56e4·9-s + 9.96e4·11-s − 6.08e4·13-s + 7.77e4·15-s − 4.34e5·17-s − 6.31e5·19-s + 1.56e6·21-s − 7.49e5·23-s − 8.81e5·25-s − 1.06e7·27-s − 7.90e6·29-s − 1.13e7·31-s − 3.22e7·33-s + 1.16e6·35-s − 1.35e7·37-s + 1.97e7·39-s − 1.88e7·41-s − 1.41e7·43-s − 1.57e7·45-s + 3.77e7·47-s − 6.42e7·49-s + 1.40e8·51-s + 1.15e8·53-s − 2.39e7·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.171·5-s − 0.761·7-s + 10/3·9-s + 2.05·11-s − 0.590·13-s + 0.396·15-s − 1.26·17-s − 1.11·19-s + 1.75·21-s − 0.558·23-s − 0.451·25-s − 3.84·27-s − 2.07·29-s − 2.20·31-s − 4.73·33-s + 0.130·35-s − 1.19·37-s + 1.36·39-s − 1.04·41-s − 0.632·43-s − 0.572·45-s + 1.12·47-s − 1.59·49-s + 2.91·51-s + 2.00·53-s − 0.352·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{28} \cdot 3^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.52994\times 10^{9}\) |
| Root analytic conductor: | \(14.0632\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 2 | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 48 p T + 939284 T^{2} - 236450096 p T^{3} - 167112133242 p^{2} T^{4} - 236450096 p^{10} T^{5} + 939284 p^{18} T^{6} + 48 p^{28} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 4840 T + 12530764 p T^{2} + 12761367208 p^{2} T^{3} + 11457097294378 p^{3} T^{4} + 12761367208 p^{11} T^{5} + 12530764 p^{19} T^{6} + 4840 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 99664 T + 10589400716 T^{2} - 699518969642576 T^{3} + 38813237732845803926 T^{4} - 699518969642576 p^{9} T^{5} + 10589400716 p^{18} T^{6} - 99664 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 360 p^{2} T + 24714128908 T^{2} + 695713226418488 T^{3} + \)\(29\!\cdots\!86\)\( T^{4} + 695713226418488 p^{9} T^{5} + 24714128908 p^{18} T^{6} + 360 p^{29} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 434952 T + 394190258396 T^{2} + 122601307335261368 T^{3} + \)\(64\!\cdots\!26\)\( T^{4} + 122601307335261368 p^{9} T^{5} + 394190258396 p^{18} T^{6} + 434952 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 631776 T + 1164722951116 T^{2} + 471204029388957920 T^{3} + \)\(52\!\cdots\!86\)\( T^{4} + 471204029388957920 p^{9} T^{5} + 1164722951116 p^{18} T^{6} + 631776 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 749392 T + 3347296061756 T^{2} + 4837140337367041424 T^{3} + \)\(64\!\cdots\!18\)\( T^{4} + 4837140337367041424 p^{9} T^{5} + 3347296061756 p^{18} T^{6} + 749392 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 7908544 T + 58432091245844 T^{2} + \)\(22\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} + \)\(22\!\cdots\!08\)\( p^{9} T^{5} + 58432091245844 p^{18} T^{6} + 7908544 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 11351240 T + 115399090325812 T^{2} + \)\(71\!\cdots\!24\)\( T^{3} + \)\(43\!\cdots\!22\)\( T^{4} + \)\(71\!\cdots\!24\)\( p^{9} T^{5} + 115399090325812 p^{18} T^{6} + 11351240 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 13592920 T + 213368728836076 T^{2} - 87591136825729972088 T^{3} + \)\(27\!\cdots\!94\)\( T^{4} - 87591136825729972088 p^{9} T^{5} + 213368728836076 p^{18} T^{6} + 13592920 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 18838888 T + 656876228382716 T^{2} + \)\(15\!\cdots\!88\)\( T^{3} + \)\(73\!\cdots\!90\)\( p T^{4} + \)\(15\!\cdots\!88\)\( p^{9} T^{5} + 656876228382716 p^{18} T^{6} + 18838888 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 14177920 T + 1417437436848556 T^{2} + \)\(23\!\cdots\!96\)\( T^{3} + \)\(93\!\cdots\!54\)\( T^{4} + \)\(23\!\cdots\!96\)\( p^{9} T^{5} + 1417437436848556 p^{18} T^{6} + 14177920 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 37779120 T + 3062567184137372 T^{2} - \)\(92\!\cdots\!16\)\( T^{3} + \)\(49\!\cdots\!30\)\( T^{4} - \)\(92\!\cdots\!16\)\( p^{9} T^{5} + 3062567184137372 p^{18} T^{6} - 37779120 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 115336512 T + 10787477606282612 T^{2} - \)\(79\!\cdots\!64\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} - \)\(79\!\cdots\!64\)\( p^{9} T^{5} + 10787477606282612 p^{18} T^{6} - 115336512 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 115028080 T + 604137837099172 p T^{2} - \)\(28\!\cdots\!56\)\( T^{3} + \)\(46\!\cdots\!30\)\( T^{4} - \)\(28\!\cdots\!56\)\( p^{9} T^{5} + 604137837099172 p^{19} T^{6} - 115028080 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 173228648 T + 46925341200786892 T^{2} - \)\(58\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!42\)\( T^{4} - \)\(58\!\cdots\!20\)\( p^{9} T^{5} + 46925341200786892 p^{18} T^{6} - 173228648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 231785104 T + 47923090341550828 T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!82\)\( T^{4} - \)\(16\!\cdots\!56\)\( p^{9} T^{5} + 47923090341550828 p^{18} T^{6} - 231785104 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 197476208 T + 67580246245585148 T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!06\)\( T^{4} + \)\(16\!\cdots\!80\)\( p^{9} T^{5} + 67580246245585148 p^{18} T^{6} + 197476208 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 44629400 T + 155814395021474812 T^{2} - \)\(40\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!82\)\( T^{4} - \)\(40\!\cdots\!56\)\( p^{9} T^{5} + 155814395021474812 p^{18} T^{6} - 44629400 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 355774584 T + 431793154567003252 T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!20\)\( p^{9} T^{5} + 431793154567003252 p^{18} T^{6} - 355774584 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 607613328 T + 689233408599137708 T^{2} - \)\(31\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!34\)\( T^{4} - \)\(31\!\cdots\!68\)\( p^{9} T^{5} + 689233408599137708 p^{18} T^{6} - 607613328 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 1157146424 T + 1248537877761010940 T^{2} + \)\(74\!\cdots\!24\)\( T^{3} + \)\(52\!\cdots\!78\)\( T^{4} + \)\(74\!\cdots\!24\)\( p^{9} T^{5} + 1248537877761010940 p^{18} T^{6} + 1157146424 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1599536472 T + 3875436244314154204 T^{2} + \)\(37\!\cdots\!12\)\( T^{3} + \)\(47\!\cdots\!58\)\( T^{4} + \)\(37\!\cdots\!12\)\( p^{9} T^{5} + 3875436244314154204 p^{18} T^{6} + 1599536472 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.00683657957516490804869633459, −6.83331583955770401612041014757, −6.56150286700787879279262862227, −6.45252073075186216882322148654, −6.43098806592512176530586382155, −5.65538336222414620798707818808, −5.57675965078564055828869114528, −5.57056375602021555588381280356, −5.54220025260477530070923431989, −4.85941400590048199661755644279, −4.62614567111465600410455426588, −4.56037079178593470546222035876, −4.05847495286327277627700126330, −3.87725505455408224177091898797, −3.67205166572386909415821595529, −3.56675181638813698086904021299, −3.41582010882243792800740163557, −2.53322926745318952162631759982, −2.24747269121811233673702362298, −2.10162043418421253309306629484, −1.94498985448749188550843302582, −1.49307994934051871180532990475, −1.19727674230204113295497530717, −1.07094905498526588778991153290, −0.61949670645190192820037766342, 0, 0, 0, 0, 0.61949670645190192820037766342, 1.07094905498526588778991153290, 1.19727674230204113295497530717, 1.49307994934051871180532990475, 1.94498985448749188550843302582, 2.10162043418421253309306629484, 2.24747269121811233673702362298, 2.53322926745318952162631759982, 3.41582010882243792800740163557, 3.56675181638813698086904021299, 3.67205166572386909415821595529, 3.87725505455408224177091898797, 4.05847495286327277627700126330, 4.56037079178593470546222035876, 4.62614567111465600410455426588, 4.85941400590048199661755644279, 5.54220025260477530070923431989, 5.57056375602021555588381280356, 5.57675965078564055828869114528, 5.65538336222414620798707818808, 6.43098806592512176530586382155, 6.45252073075186216882322148654, 6.56150286700787879279262862227, 6.83331583955770401612041014757, 7.00683657957516490804869633459