Dirichlet series
| L(s) = 1 | − 46·2-s − 972·3-s − 714·4-s + 4.47e4·6-s − 6.83e4·7-s − 3.20e4·8-s + 5.90e5·9-s − 9.99e4·11-s + 6.94e5·12-s − 2.30e6·13-s + 3.14e6·14-s + 3.88e6·16-s + 3.44e6·17-s − 2.71e7·18-s − 4.21e6·19-s + 6.64e7·21-s + 4.59e6·22-s − 5.26e7·23-s + 3.11e7·24-s + 1.06e8·26-s − 2.86e8·27-s + 4.88e7·28-s + 2.17e8·29-s − 3.26e8·31-s + 4.65e7·32-s + 9.71e7·33-s − 1.58e8·34-s + ⋯ |
| L(s) = 1 | − 1.01·2-s − 2.30·3-s − 0.348·4-s + 2.34·6-s − 1.53·7-s − 0.345·8-s + 10/3·9-s − 0.187·11-s + 0.805·12-s − 1.72·13-s + 1.56·14-s + 0.926·16-s + 0.588·17-s − 3.38·18-s − 0.390·19-s + 3.55·21-s + 0.190·22-s − 1.70·23-s + 0.798·24-s + 1.75·26-s − 3.84·27-s + 0.536·28-s + 1.96·29-s − 2.04·31-s + 0.245·32-s + 0.432·33-s − 0.597·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(31640625\) = \(3^{4} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.10272\times 10^{7}\) |
| Root analytic conductor: | \(7.59116\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 31640625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(0.06160130698\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06160130698\) |
| \(L(\frac{13}{2})\) | 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 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) |
| 5 | \( 1 \) | ||
| good | 2 | $C_2 \wr S_4$ | \( 1 + 23 p T + 1415 p T^{2} + 12191 p^{4} T^{3} + 134099 p^{6} T^{4} + 12191 p^{15} T^{5} + 1415 p^{23} T^{6} + 23 p^{34} T^{7} + p^{44} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 68372 T + 3685360978 T^{2} + 22270114394664 p T^{3} + 185231300643789915 p^{2} T^{4} + 22270114394664 p^{12} T^{5} + 3685360978 p^{22} T^{6} + 68372 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 99944 T + 3829207972 p^{2} T^{2} + 53580549386883400 T^{3} + \)\(17\!\cdots\!86\)\( T^{4} + 53580549386883400 p^{11} T^{5} + 3829207972 p^{24} T^{6} + 99944 p^{33} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 2306276 T + 7411071016186 T^{2} + 11446025290832742208 T^{3} + \)\(20\!\cdots\!35\)\( T^{4} + 11446025290832742208 p^{11} T^{5} + 7411071016186 p^{22} T^{6} + 2306276 p^{33} T^{7} + p^{44} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 3443816 T + 102603037682428 T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} - \)\(36\!\cdots\!40\)\( p^{11} T^{5} + 102603037682428 p^{22} T^{6} - 3443816 p^{33} T^{7} + p^{44} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 4214548 T - 3406227372638 T^{2} - \)\(10\!\cdots\!24\)\( T^{3} + \)\(15\!\cdots\!59\)\( T^{4} - \)\(10\!\cdots\!24\)\( p^{11} T^{5} - 3406227372638 p^{22} T^{6} + 4214548 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 52691304 T + 4586182697931092 T^{2} + \)\(15\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!10\)\( T^{4} + \)\(15\!\cdots\!84\)\( p^{11} T^{5} + 4586182697931092 p^{22} T^{6} + 52691304 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 217393304 T + 48076892013700300 T^{2} - \)\(67\!\cdots\!08\)\( T^{3} + \)\(90\!\cdots\!18\)\( T^{4} - \)\(67\!\cdots\!08\)\( p^{11} T^{5} + 48076892013700300 p^{22} T^{6} - 217393304 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 10526356 p T + 110894376843952498 T^{2} + \)\(21\!\cdots\!48\)\( T^{3} + \)\(42\!\cdots\!79\)\( T^{4} + \)\(21\!\cdots\!48\)\( p^{11} T^{5} + 110894376843952498 p^{22} T^{6} + 10526356 p^{34} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 273252872 T - 81057457446333812 T^{2} + \)\(33\!\cdots\!48\)\( T^{3} + \)\(69\!\cdots\!50\)\( T^{4} + \)\(33\!\cdots\!48\)\( p^{11} T^{5} - 81057457446333812 p^{22} T^{6} + 273252872 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 22069456 T + 796406960614594468 T^{2} + \)\(47\!\cdots\!48\)\( T^{3} + \)\(39\!\cdots\!14\)\( T^{4} + \)\(47\!\cdots\!48\)\( p^{11} T^{5} + 796406960614594468 p^{22} T^{6} + 22069456 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 705091900 T + 3103662515659227250 T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!23\)\( T^{4} - \)\(18\!\cdots\!00\)\( p^{11} T^{5} + 3103662515659227250 p^{22} T^{6} - 705091900 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 187768360 T + 7352359892990648980 T^{2} + \)\(74\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!18\)\( T^{4} + \)\(74\!\cdots\!80\)\( p^{11} T^{5} + 7352359892990648980 p^{22} T^{6} + 187768360 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 6392224256 T + 955975989160217924 p T^{2} - \)\(18\!\cdots\!96\)\( T^{3} + \)\(76\!\cdots\!50\)\( T^{4} - \)\(18\!\cdots\!96\)\( p^{11} T^{5} + 955975989160217924 p^{23} T^{6} - 6392224256 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 36710008 T + 45110259195200964772 T^{2} + \)\(24\!\cdots\!04\)\( T^{3} + \)\(77\!\cdots\!34\)\( T^{4} + \)\(24\!\cdots\!04\)\( p^{11} T^{5} + 45110259195200964772 p^{22} T^{6} - 36710008 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 11538870620 T + \)\(18\!\cdots\!46\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!71\)\( T^{4} - \)\(14\!\cdots\!80\)\( p^{11} T^{5} + \)\(18\!\cdots\!46\)\( p^{22} T^{6} - 11538870620 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 37721158484 T + \)\(95\!\cdots\!78\)\( T^{2} + \)\(15\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!51\)\( T^{4} + \)\(15\!\cdots\!60\)\( p^{11} T^{5} + \)\(95\!\cdots\!78\)\( p^{22} T^{6} + 37721158484 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 8211316688 T + \)\(77\!\cdots\!88\)\( T^{2} - \)\(53\!\cdots\!36\)\( T^{3} + \)\(25\!\cdots\!70\)\( T^{4} - \)\(53\!\cdots\!36\)\( p^{11} T^{5} + \)\(77\!\cdots\!88\)\( p^{22} T^{6} - 8211316688 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 5713413224 T + \)\(85\!\cdots\!72\)\( T^{2} + \)\(92\!\cdots\!84\)\( T^{3} + \)\(32\!\cdots\!30\)\( T^{4} + \)\(92\!\cdots\!84\)\( p^{11} T^{5} + \)\(85\!\cdots\!72\)\( p^{22} T^{6} + 5713413224 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 45026381600 T + \)\(17\!\cdots\!16\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!46\)\( T^{4} - \)\(23\!\cdots\!00\)\( p^{11} T^{5} + \)\(17\!\cdots\!16\)\( p^{22} T^{6} - 45026381600 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 104211315528 T + \)\(88\!\cdots\!40\)\( T^{2} - \)\(45\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!26\)\( T^{4} - \)\(45\!\cdots\!92\)\( p^{11} T^{5} + \)\(88\!\cdots\!40\)\( p^{22} T^{6} - 104211315528 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 111829609152 T + \)\(12\!\cdots\!52\)\( T^{2} - \)\(81\!\cdots\!04\)\( T^{3} + \)\(51\!\cdots\!14\)\( T^{4} - \)\(81\!\cdots\!04\)\( p^{11} T^{5} + \)\(12\!\cdots\!52\)\( p^{22} T^{6} - 111829609152 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 77104304804 T + \)\(13\!\cdots\!18\)\( T^{2} + \)\(68\!\cdots\!20\)\( p T^{3} + \)\(10\!\cdots\!91\)\( T^{4} + \)\(68\!\cdots\!20\)\( p^{12} T^{5} + \)\(13\!\cdots\!18\)\( p^{22} T^{6} + 77104304804 p^{33} T^{7} + p^{44} T^{8} \) | |
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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
−8.721998570217715619608792946391, −7.84788773902837325224986061916, −7.76685837101328171342722659810, −7.73498407435062557038314749540, −7.03382292092208553466015778437, −6.79346174913238896384274335564, −6.60992869808452143246110495174, −6.24757457635675262259273689723, −5.96598829892072457143370950505, −5.78616905092745413821780048668, −5.14106200929426987481085654084, −5.10122291081021633418443756008, −5.02259240664414017448870264265, −4.20653268208046993148959506026, −3.90601471896478904359485772861, −3.85790151846453960544671184293, −3.15803354950337890640174599991, −2.82984589083576600879334618289, −2.39185091644328229119199953264, −1.81140754990215183256803146741, −1.75416452718242580286976952122, −0.67396815517567529183748950542, −0.63468406291854998280705550143, −0.59907134740720459412962527282, −0.10662929873570255012677026839, 0.10662929873570255012677026839, 0.59907134740720459412962527282, 0.63468406291854998280705550143, 0.67396815517567529183748950542, 1.75416452718242580286976952122, 1.81140754990215183256803146741, 2.39185091644328229119199953264, 2.82984589083576600879334618289, 3.15803354950337890640174599991, 3.85790151846453960544671184293, 3.90601471896478904359485772861, 4.20653268208046993148959506026, 5.02259240664414017448870264265, 5.10122291081021633418443756008, 5.14106200929426987481085654084, 5.78616905092745413821780048668, 5.96598829892072457143370950505, 6.24757457635675262259273689723, 6.60992869808452143246110495174, 6.79346174913238896384274335564, 7.03382292092208553466015778437, 7.73498407435062557038314749540, 7.76685837101328171342722659810, 7.84788773902837325224986061916, 8.721998570217715619608792946391