Dirichlet series
| L(s) = 1 | − 33·2-s − 2.62e4·3-s − 4.19e4·4-s + 1.56e6·5-s + 8.66e5·6-s + 1.75e7·7-s + 2.40e7·8-s + 4.30e8·9-s − 5.15e7·10-s − 5.75e8·11-s + 1.10e9·12-s − 5.04e9·13-s − 5.80e8·14-s − 4.10e10·15-s − 1.23e10·16-s + 4.57e10·17-s − 1.42e10·18-s + 1.98e11·19-s − 6.55e10·20-s − 4.61e11·21-s + 1.89e10·22-s + 9.61e10·23-s − 6.31e11·24-s + 1.52e12·25-s + 1.66e11·26-s − 5.64e12·27-s − 7.37e11·28-s + ⋯ |
| L(s) = 1 | − 0.0911·2-s − 2.30·3-s − 0.319·4-s + 1.78·5-s + 0.210·6-s + 1.15·7-s + 0.506·8-s + 10/3·9-s − 0.163·10-s − 0.809·11-s + 0.738·12-s − 1.71·13-s − 0.105·14-s − 4.13·15-s − 0.721·16-s + 1.59·17-s − 0.303·18-s + 2.68·19-s − 0.572·20-s − 2.66·21-s + 0.0737·22-s + 0.255·23-s − 1.17·24-s + 2·25-s + 0.156·26-s − 3.84·27-s − 0.368·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(50625\) = \(3^{4} \cdot 5^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(570527.\) |
| Root analytic conductor: | \(5.24245\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 50625,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
Particular Values
| \(L(9)\) | \(\approx\) | \(3.328753997\) |
| \(L(\frac12)\) | \(\approx\) | \(3.328753997\) |
| \(L(\frac{19}{2})\) | 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 | $C_1$ | \( ( 1 + p^{8} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - p^{8} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 33 T + 21505 p T^{2} - 332057 p^{6} T^{3} + 24796413 p^{9} T^{4} - 332057 p^{23} T^{5} + 21505 p^{35} T^{6} + 33 p^{51} T^{7} + p^{68} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 2511872 p T + 477041201386012 T^{2} - \)\(15\!\cdots\!56\)\( p^{2} T^{3} + \)\(66\!\cdots\!10\)\( p^{5} T^{4} - \)\(15\!\cdots\!56\)\( p^{19} T^{5} + 477041201386012 p^{34} T^{6} - 2511872 p^{52} T^{7} + p^{68} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 52317744 p T - 514261511222548 p^{2} T^{2} + \)\(11\!\cdots\!40\)\( p^{3} T^{3} + \)\(30\!\cdots\!26\)\( p^{4} T^{4} + \)\(11\!\cdots\!40\)\( p^{20} T^{5} - 514261511222548 p^{36} T^{6} + 52317744 p^{52} T^{7} + p^{68} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 5049645832 T + 2326205135243261308 p T^{2} + \)\(59\!\cdots\!96\)\( p^{2} T^{3} + \)\(17\!\cdots\!50\)\( p^{3} T^{4} + \)\(59\!\cdots\!96\)\( p^{19} T^{5} + 2326205135243261308 p^{35} T^{6} + 5049645832 p^{51} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 45757603848 T + \)\(15\!\cdots\!72\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(36\!\cdots\!00\)\( p^{17} T^{5} + \)\(15\!\cdots\!72\)\( p^{34} T^{6} - 45757603848 p^{51} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 198913764368 T + 66803001199960143052 p^{2} T^{2} - \)\(18\!\cdots\!96\)\( T^{3} + \)\(14\!\cdots\!74\)\( T^{4} - \)\(18\!\cdots\!96\)\( p^{17} T^{5} + 66803001199960143052 p^{36} T^{6} - 198913764368 p^{51} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 96105042432 T + \)\(25\!\cdots\!08\)\( T^{2} - \)\(22\!\cdots\!28\)\( T^{3} + \)\(43\!\cdots\!30\)\( T^{4} - \)\(22\!\cdots\!28\)\( p^{17} T^{5} + \)\(25\!\cdots\!08\)\( p^{34} T^{6} - 96105042432 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 4037480662776 T + \)\(23\!\cdots\!00\)\( T^{2} - \)\(70\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!38\)\( T^{4} - \)\(70\!\cdots\!12\)\( p^{17} T^{5} + \)\(23\!\cdots\!00\)\( p^{34} T^{6} - 4037480662776 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 14684104369664 T + \)\(14\!\cdots\!68\)\( T^{2} - \)\(98\!\cdots\!12\)\( T^{3} + \)\(54\!\cdots\!54\)\( T^{4} - \)\(98\!\cdots\!12\)\( p^{17} T^{5} + \)\(14\!\cdots\!68\)\( p^{34} T^{6} - 14684104369664 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 3157207163176 T + \)\(38\!\cdots\!72\)\( T^{2} - \)\(56\!\cdots\!44\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} - \)\(56\!\cdots\!44\)\( p^{17} T^{5} + \)\(38\!\cdots\!72\)\( p^{34} T^{6} + 3157207163176 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 1165979047464 T + \)\(60\!\cdots\!48\)\( T^{2} + \)\(10\!\cdots\!28\)\( p T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(10\!\cdots\!28\)\( p^{18} T^{5} + \)\(60\!\cdots\!48\)\( p^{34} T^{6} - 1165979047464 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 169312174332400 T + \)\(31\!\cdots\!60\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!98\)\( T^{4} + \)\(31\!\cdots\!00\)\( p^{17} T^{5} + \)\(31\!\cdots\!60\)\( p^{34} T^{6} + 169312174332400 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 155677669862880 T + \)\(55\!\cdots\!20\)\( T^{2} + \)\(55\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!38\)\( T^{4} + \)\(55\!\cdots\!60\)\( p^{17} T^{5} + \)\(55\!\cdots\!20\)\( p^{34} T^{6} + 155677669862880 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 1177003151760168 T + \)\(72\!\cdots\!28\)\( T^{2} + \)\(35\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!90\)\( T^{4} + \)\(35\!\cdots\!32\)\( p^{17} T^{5} + \)\(72\!\cdots\!28\)\( p^{34} T^{6} + 1177003151760168 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 961128761787408 T + \)\(39\!\cdots\!92\)\( T^{2} + \)\(25\!\cdots\!56\)\( T^{3} + \)\(65\!\cdots\!74\)\( T^{4} + \)\(25\!\cdots\!56\)\( p^{17} T^{5} + \)\(39\!\cdots\!92\)\( p^{34} T^{6} + 961128761787408 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 1266269141333240 T + \)\(55\!\cdots\!16\)\( T^{2} - \)\(59\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!46\)\( T^{4} - \)\(59\!\cdots\!60\)\( p^{17} T^{5} + \)\(55\!\cdots\!16\)\( p^{34} T^{6} - 1266269141333240 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 3378394923251152 T + \)\(60\!\cdots\!72\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} - \)\(11\!\cdots\!34\)\( T^{4} - \)\(37\!\cdots\!00\)\( p^{17} T^{5} + \)\(60\!\cdots\!72\)\( p^{34} T^{6} + 3378394923251152 p^{51} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 13149094845793248 T + \)\(71\!\cdots\!28\)\( T^{2} - \)\(38\!\cdots\!16\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} - \)\(38\!\cdots\!16\)\( p^{17} T^{5} + \)\(71\!\cdots\!28\)\( p^{34} T^{6} - 13149094845793248 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 19279336942377512 T + \)\(28\!\cdots\!88\)\( T^{2} - \)\(28\!\cdots\!48\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!48\)\( p^{17} T^{5} + \)\(28\!\cdots\!88\)\( p^{34} T^{6} - 19279336942377512 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 3966417576450560 T + \)\(50\!\cdots\!36\)\( T^{2} + \)\(75\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!86\)\( T^{4} + \)\(75\!\cdots\!80\)\( p^{17} T^{5} + \)\(50\!\cdots\!36\)\( p^{34} T^{6} - 3966417576450560 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 2071402099563984 T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(24\!\cdots\!84\)\( T^{3} + \)\(99\!\cdots\!86\)\( T^{4} + \)\(24\!\cdots\!84\)\( p^{17} T^{5} + \)\(16\!\cdots\!20\)\( p^{34} T^{6} + 2071402099563984 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 29070369191797272 T + \)\(51\!\cdots\!72\)\( T^{2} + \)\(11\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!94\)\( T^{4} + \)\(11\!\cdots\!24\)\( p^{17} T^{5} + \)\(51\!\cdots\!72\)\( p^{34} T^{6} + 29070369191797272 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 122163393367080968 T + \)\(27\!\cdots\!32\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!26\)\( T^{4} - \)\(22\!\cdots\!00\)\( p^{17} T^{5} + \)\(27\!\cdots\!32\)\( p^{34} T^{6} - 122163393367080968 p^{51} T^{7} + p^{68} T^{8} \) | |
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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
−10.29093331536883434535754987926, −9.983255526068871510643920138242, −9.885388796051412657939610952719, −9.677981153384047413910130812298, −9.410470938397379270224694796583, −8.202784185795104819546644356709, −8.103542866648559399312839866327, −7.84332697294508127319960709725, −7.10173802901433132419318993940, −6.80052792081051672867616520788, −6.53053439602377322099547749612, −5.89493980222124206734806630661, −5.68510269826782471605010645561, −4.98509729739291771200423340369, −4.86806071729717974474124262359, −4.81821226943203885461277813109, −4.75979629477430724444112079692, −3.28192418132394583377725019418, −3.09072603208374605279806712364, −2.41901204777892183785058960111, −1.84046120303542545290048268367, −1.49889297665907234001738396594, −0.955166660030715406991007639625, −0.900611194307495217666167412874, −0.36325077466978849698547560827, 0.36325077466978849698547560827, 0.900611194307495217666167412874, 0.955166660030715406991007639625, 1.49889297665907234001738396594, 1.84046120303542545290048268367, 2.41901204777892183785058960111, 3.09072603208374605279806712364, 3.28192418132394583377725019418, 4.75979629477430724444112079692, 4.81821226943203885461277813109, 4.86806071729717974474124262359, 4.98509729739291771200423340369, 5.68510269826782471605010645561, 5.89493980222124206734806630661, 6.53053439602377322099547749612, 6.80052792081051672867616520788, 7.10173802901433132419318993940, 7.84332697294508127319960709725, 8.103542866648559399312839866327, 8.202784185795104819546644356709, 9.410470938397379270224694796583, 9.677981153384047413910130812298, 9.885388796051412657939610952719, 9.983255526068871510643920138242, 10.29093331536883434535754987926