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
| \(L(5)\) | \(\approx\) | \(55.55838116\) |
| \(L(\frac12)\) | \(\approx\) | \(55.55838116\) |
| \(L(\frac{11}{2})\) | 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
−8.849870672939370368966084802636, −8.421263390648970735605666486155, −8.280046923849814055253957666622, −8.238298619961487163949151987536, −7.69374788497463844484481740285, −7.36621639111516527951331746945, −7.29682230702268748598941651149, −6.56669842098099022391237817997, −6.55055443787165191249302385815, −6.14215255004559162602801585586, −5.43082428076890033246884032281, −5.13964563865468749886041247245, −5.05727587777490986030841672428, −4.23581304620465644697533075921, −4.21861256085125154573703562986, −3.70902244325771290126105816804, −3.48452237788057898030931733347, −3.15351868368715009076750678657, −2.87971242301224102353796377981, −2.23210501517038047066633690797, −1.81740846936672262754247996565, −1.60749850767918153032171914673, −0.987694852144141500660473142921, −0.907986123055654896529936135141, −0.895601158618466947108564681988, 0.895601158618466947108564681988, 0.907986123055654896529936135141, 0.987694852144141500660473142921, 1.60749850767918153032171914673, 1.81740846936672262754247996565, 2.23210501517038047066633690797, 2.87971242301224102353796377981, 3.15351868368715009076750678657, 3.48452237788057898030931733347, 3.70902244325771290126105816804, 4.21861256085125154573703562986, 4.23581304620465644697533075921, 5.05727587777490986030841672428, 5.13964563865468749886041247245, 5.43082428076890033246884032281, 6.14215255004559162602801585586, 6.55055443787165191249302385815, 6.56669842098099022391237817997, 7.29682230702268748598941651149, 7.36621639111516527951331746945, 7.69374788497463844484481740285, 8.238298619961487163949151987536, 8.280046923849814055253957666622, 8.421263390648970735605666486155, 8.849870672939370368966084802636