Dirichlet series
| L(s) = 1 | + 33·2-s + 2.62e4·3-s − 4.19e4·4-s + 8.66e5·6-s − 1.75e7·7-s − 2.40e7·8-s + 4.30e8·9-s − 5.75e8·11-s − 1.10e9·12-s + 5.04e9·13-s − 5.80e8·14-s − 1.23e10·16-s − 4.57e10·17-s + 1.42e10·18-s + 1.98e11·19-s − 4.61e11·21-s − 1.89e10·22-s − 9.61e10·23-s − 6.31e11·24-s + 1.66e11·26-s + 5.64e12·27-s + 7.37e11·28-s + 4.03e12·29-s + 1.46e13·31-s − 1.68e12·32-s − 1.51e13·33-s − 1.51e12·34-s + ⋯ |
| L(s) = 1 | + 0.0911·2-s + 2.30·3-s − 0.319·4-s + 0.210·6-s − 1.15·7-s − 0.506·8-s + 10/3·9-s − 0.809·11-s − 0.738·12-s + 1.71·13-s − 0.105·14-s − 0.721·16-s − 1.59·17-s + 0.303·18-s + 2.68·19-s − 2.66·21-s − 0.0737·22-s − 0.255·23-s − 1.17·24-s + 0.156·26-s + 3.84·27-s + 0.368·28-s + 1.49·29-s + 3.09·31-s − 0.271·32-s − 1.86·33-s − 0.145·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(31640625\) = \(3^{4} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.56579\times 10^{8}\) |
| Root analytic conductor: | \(11.7224\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 31640625,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
Particular Values
| \(L(9)\) | \(\approx\) | \(20.02178711\) |
| \(L(\frac12)\) | \(\approx\) | \(20.02178711\) |
| \(L(\frac{19}{2})\) | 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 | \( 1 \) | ||
| 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
−7.46662467541437810344778312139, −7.29074086122433977660262783926, −7.02220333147739756895333145897, −6.53765259997018010764538682900, −6.42806629322770642135224989071, −6.32111168721883479584543711661, −5.60691423272265285423956738337, −5.48859378724785992656446333292, −5.18697815873787855485825870527, −4.53125544430917296830210670867, −4.36997376907068022087247114398, −4.07348983712566437212251866623, −4.06190653216132608293161404305, −3.34958893132719430117956623921, −3.28512724572824885058823049643, −2.97699136068393620573479294716, −2.68626139660136395917519583238, −2.63951845883758872163908654135, −2.27632471893138818590397523040, −1.89855003971937103992837075310, −1.50113858772396603375961197599, −0.916790420248911209550592051992, −0.854608537436753240033660776621, −0.790984129538614133476993147546, −0.29184656463571752395965434864, 0.29184656463571752395965434864, 0.790984129538614133476993147546, 0.854608537436753240033660776621, 0.916790420248911209550592051992, 1.50113858772396603375961197599, 1.89855003971937103992837075310, 2.27632471893138818590397523040, 2.63951845883758872163908654135, 2.68626139660136395917519583238, 2.97699136068393620573479294716, 3.28512724572824885058823049643, 3.34958893132719430117956623921, 4.06190653216132608293161404305, 4.07348983712566437212251866623, 4.36997376907068022087247114398, 4.53125544430917296830210670867, 5.18697815873787855485825870527, 5.48859378724785992656446333292, 5.60691423272265285423956738337, 6.32111168721883479584543711661, 6.42806629322770642135224989071, 6.53765259997018010764538682900, 7.02220333147739756895333145897, 7.29074086122433977660262783926, 7.46662467541437810344778312139