Dirichlet series
| L(s) = 1 | + 2·2-s − 324·3-s − 126·4-s − 648·6-s − 1.30e4·7-s + 7.61e3·8-s + 6.56e4·9-s + 1.04e5·11-s + 4.08e4·12-s − 1.40e5·13-s − 2.60e4·14-s + 1.24e5·16-s − 4.89e5·17-s + 1.31e5·18-s + 3.93e5·19-s + 4.22e6·21-s + 2.09e5·22-s − 2.32e6·23-s − 2.46e6·24-s − 2.81e5·26-s − 1.06e7·27-s + 1.64e6·28-s + 4.92e6·29-s − 9.75e4·31-s + 2.54e6·32-s − 3.39e7·33-s − 9.78e5·34-s + ⋯ |
| L(s) = 1 | + 0.0883·2-s − 2.30·3-s − 0.246·4-s − 0.204·6-s − 2.05·7-s + 0.657·8-s + 10/3·9-s + 2.15·11-s + 0.568·12-s − 1.36·13-s − 0.181·14-s + 0.473·16-s − 1.42·17-s + 0.294·18-s + 0.692·19-s + 4.73·21-s + 0.190·22-s − 1.73·23-s − 1.51·24-s − 0.120·26-s − 3.84·27-s + 0.505·28-s + 1.29·29-s − 0.0189·31-s + 0.429·32-s − 4.97·33-s − 0.125·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(31640625\) = \(3^{4} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.22635\times 10^{6}\) |
| Root analytic conductor: | \(6.21511\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 31640625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(1.707637514\) |
| \(L(\frac12)\) | \(\approx\) | \(1.707637514\) |
| \(L(\frac{11}{2})\) | 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 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
| 5 | \( 1 \) | ||
| good | 2 | $C_2 \wr C_2\wr C_2$ | \( 1 - p T + 65 p T^{2} - 127 p^{6} T^{3} - 2387 p^{5} T^{4} - 127 p^{15} T^{5} + 65 p^{19} T^{6} - p^{28} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 13036 T + 125326402 T^{2} + 104414940408 p T^{3} + 94875004775595 p^{2} T^{4} + 104414940408 p^{10} T^{5} + 125326402 p^{18} T^{6} + 13036 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 - 104696 T + 8215985332 T^{2} - 516596761565080 T^{3} + 29503741825296604886 T^{4} - 516596761565080 p^{9} T^{5} + 8215985332 p^{18} T^{6} - 104696 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 140812 T + 17419184074 T^{2} + 1151392278946304 T^{3} + 51896580315633752675 T^{4} + 1151392278946304 p^{9} T^{5} + 17419184074 p^{18} T^{6} + 140812 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 489352 T + 316005336652 T^{2} + 70146651286188920 T^{3} + \)\(35\!\cdots\!06\)\( T^{4} + 70146651286188920 p^{9} T^{5} + 316005336652 p^{18} T^{6} + 489352 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - 393308 T + 721294411042 T^{2} - 280833919842462136 T^{3} + \)\(33\!\cdots\!79\)\( T^{4} - 280833919842462136 p^{9} T^{5} + 721294411042 p^{18} T^{6} - 393308 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2326488 T + 4936689565988 T^{2} + 6431711107977715032 T^{3} + \)\(95\!\cdots\!50\)\( T^{4} + 6431711107977715032 p^{9} T^{5} + 4936689565988 p^{18} T^{6} + 2326488 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4926616 T + 28214213209660 T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + \)\(57\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!72\)\( p^{9} T^{5} + 28214213209660 p^{18} T^{6} - 4926616 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 97516 T + 33448054683298 T^{2} + 47518669873090400808 T^{3} + \)\(13\!\cdots\!59\)\( T^{4} + 47518669873090400808 p^{9} T^{5} + 33448054683298 p^{18} T^{6} + 97516 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4958984 T + 392698806874732 T^{2} - \)\(21\!\cdots\!24\)\( T^{3} + \)\(69\!\cdots\!70\)\( T^{4} - \)\(21\!\cdots\!24\)\( p^{9} T^{5} + 392698806874732 p^{18} T^{6} - 4958984 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 996656 T + 1183182536788468 T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(55\!\cdots\!54\)\( T^{4} + \)\(16\!\cdots\!68\)\( p^{9} T^{5} + 1183182536788468 p^{18} T^{6} + 996656 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 28298860 T + 1667864367008050 T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!23\)\( T^{4} + \)\(33\!\cdots\!80\)\( p^{9} T^{5} + 1667864367008050 p^{18} T^{6} + 28298860 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 30714920 T + 3230436540060820 T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!78\)\( T^{4} - \)\(10\!\cdots\!40\)\( p^{9} T^{5} + 3230436540060820 p^{18} T^{6} - 30714920 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 1333856 p T + 2311388195091268 T^{2} - \)\(19\!\cdots\!48\)\( T^{3} - \)\(17\!\cdots\!90\)\( T^{4} - \)\(19\!\cdots\!48\)\( p^{9} T^{5} + 2311388195091268 p^{18} T^{6} + 1333856 p^{28} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 225946712 T + 48843492591377092 T^{2} - \)\(60\!\cdots\!04\)\( T^{3} + \)\(69\!\cdots\!34\)\( T^{4} - \)\(60\!\cdots\!04\)\( p^{9} T^{5} + 48843492591377092 p^{18} T^{6} - 225946712 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6295340 T + 24354834957761386 T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(32\!\cdots\!11\)\( T^{4} - \)\(12\!\cdots\!60\)\( p^{9} T^{5} + 24354834957761386 p^{18} T^{6} - 6295340 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 217434788 T + 97369659228201442 T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!11\)\( T^{4} - \)\(15\!\cdots\!00\)\( p^{9} T^{5} + 97369659228201442 p^{18} T^{6} - 217434788 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 22716688 T + 109989327828315628 T^{2} + \)\(30\!\cdots\!24\)\( T^{3} + \)\(62\!\cdots\!70\)\( T^{4} + \)\(30\!\cdots\!24\)\( p^{9} T^{5} + 109989327828315628 p^{18} T^{6} - 22716688 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 79864888 T + 132631813044998908 T^{2} + \)\(14\!\cdots\!12\)\( T^{3} + \)\(91\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!12\)\( p^{9} T^{5} + 132631813044998908 p^{18} T^{6} + 79864888 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 1276962080 T + 1010097812279535676 T^{2} - \)\(54\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!66\)\( T^{4} - \)\(54\!\cdots\!60\)\( p^{9} T^{5} + 1010097812279535676 p^{18} T^{6} - 1276962080 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 175482984 T + 359109872421275300 T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(66\!\cdots\!26\)\( T^{4} + \)\(15\!\cdots\!64\)\( p^{9} T^{5} + 359109872421275300 p^{18} T^{6} + 175482984 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 897754752 T + 512971838285193572 T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(82\!\cdots\!34\)\( T^{4} + \)\(11\!\cdots\!64\)\( p^{9} T^{5} + 512971838285193572 p^{18} T^{6} + 897754752 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 1702783612 T + 2787256419081357322 T^{2} + \)\(24\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!51\)\( T^{4} + \)\(24\!\cdots\!20\)\( p^{9} T^{5} + 2787256419081357322 p^{18} T^{6} + 1702783612 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−9.209064544647802223179813353535, −8.456770090774623251161446112132, −8.092168786905024436605764957049, −8.012894698346557055783844170529, −7.17949210298578787504608201370, −7.14784394229914994249257453229, −6.71928682701366791533165087327, −6.47916496601013648622604468523, −6.38313538956215118897276890568, −6.31470196760494146438799252959, −5.53400180051115665680284570210, −5.34051593524403313704600985850, −5.16520244869289175613547185809, −4.51031178775217665051779761304, −4.18628510905612335846288872180, −4.11550548530651072874817993547, −3.81338384746206622690634617591, −3.07006791295210231884821216250, −2.88838736067456928759445671145, −2.06512097027002640282853680844, −1.80246897422112428003393016928, −1.37794126608773185353939152433, −0.60930814756718010506045240485, −0.58102504743008153422170020773, −0.42093737406922370029831749621, 0.42093737406922370029831749621, 0.58102504743008153422170020773, 0.60930814756718010506045240485, 1.37794126608773185353939152433, 1.80246897422112428003393016928, 2.06512097027002640282853680844, 2.88838736067456928759445671145, 3.07006791295210231884821216250, 3.81338384746206622690634617591, 4.11550548530651072874817993547, 4.18628510905612335846288872180, 4.51031178775217665051779761304, 5.16520244869289175613547185809, 5.34051593524403313704600985850, 5.53400180051115665680284570210, 6.31470196760494146438799252959, 6.38313538956215118897276890568, 6.47916496601013648622604468523, 6.71928682701366791533165087327, 7.14784394229914994249257453229, 7.17949210298578787504608201370, 8.012894698346557055783844170529, 8.092168786905024436605764957049, 8.456770090774623251161446112132, 9.209064544647802223179813353535