Dirichlet series
L(s) = 1 | + 6.98e4·2-s − 1.71e12·4-s − 1.18e14·5-s + 1.50e17·7-s − 2.26e18·8-s − 8.27e18·10-s − 7.24e20·11-s − 8.84e21·13-s + 1.04e22·14-s − 3.10e24·16-s + 3.82e25·17-s + 2.61e26·19-s + 2.03e26·20-s − 5.06e25·22-s + 1.53e28·23-s − 2.52e28·25-s − 6.17e26·26-s − 2.58e29·28-s + 1.03e30·29-s + 9.28e30·31-s + 2.31e30·32-s + 2.66e30·34-s − 1.78e31·35-s + 2.00e32·37-s + 1.82e31·38-s + 2.68e32·40-s − 2.35e33·41-s + ⋯ |
L(s) = 1 | + 0.0470·2-s − 0.781·4-s − 0.555·5-s + 0.711·7-s − 0.694·8-s − 0.0261·10-s − 0.324·11-s − 0.129·13-s + 0.0335·14-s − 0.642·16-s + 2.28·17-s + 1.59·19-s + 0.434·20-s − 0.0152·22-s + 1.87·23-s − 0.555·25-s − 0.00607·26-s − 0.556·28-s + 1.08·29-s + 2.48·31-s + 0.322·32-s + 0.107·34-s − 0.395·35-s + 1.42·37-s + 0.0752·38-s + 0.386·40-s − 2.04·41-s + ⋯ |
Functional equation
Invariants
Degree: | \(8\) |
Conductor: | \(6561\) = \(3^{8}\) |
Sign: | $1$ |
Analytic conductor: | \(8.43152\times 10^{7}\) |
Root analytic conductor: | \(9.78899\) |
Motivic weight: | \(41\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((8,\ 6561,\ (\ :41/2, 41/2, 41/2, 41/2),\ 1)\) |
Particular Values
\(L(21)\) | \(\approx\) | \(11.13895273\) |
\(L(\frac12)\) | \(\approx\) | \(11.13895273\) |
\(L(\frac{43}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 3 | \( 1 \) | |
good | 2 | $C_2 \wr S_4$ | \( 1 - 34911 p T + 53875321015 p^{5} T^{2} + 247085990567271 p^{13} T^{3} + 171951681772402383 p^{25} T^{4} + 247085990567271 p^{54} T^{5} + 53875321015 p^{87} T^{6} - 34911 p^{124} T^{7} + p^{164} T^{8} \) |
5 | $C_2 \wr S_4$ | \( 1 + 23707392755256 p T + \)\(31\!\cdots\!68\)\( p^{3} T^{2} + \)\(16\!\cdots\!68\)\( p^{7} T^{3} + \)\(21\!\cdots\!46\)\( p^{13} T^{4} + \)\(16\!\cdots\!68\)\( p^{48} T^{5} + \)\(31\!\cdots\!68\)\( p^{85} T^{6} + 23707392755256 p^{124} T^{7} + p^{164} T^{8} \) | |
7 | $C_2 \wr S_4$ | \( 1 - 150256264888927136 T + \)\(23\!\cdots\!48\)\( p^{2} T^{2} - \)\(55\!\cdots\!56\)\( p^{4} T^{3} + \)\(17\!\cdots\!90\)\( p^{9} T^{4} - \)\(55\!\cdots\!56\)\( p^{45} T^{5} + \)\(23\!\cdots\!48\)\( p^{84} T^{6} - 150256264888927136 p^{123} T^{7} + p^{164} T^{8} \) | |
11 | $C_2 \wr S_4$ | \( 1 + 65891877021040062096 p T + \)\(18\!\cdots\!32\)\( p^{2} T^{2} + \)\(63\!\cdots\!80\)\( p^{5} T^{3} + \)\(16\!\cdots\!46\)\( p^{8} T^{4} + \)\(63\!\cdots\!80\)\( p^{46} T^{5} + \)\(18\!\cdots\!32\)\( p^{84} T^{6} + 65891877021040062096 p^{124} T^{7} + p^{164} T^{8} \) | |
13 | $C_2 \wr S_4$ | \( 1 + \)\(88\!\cdots\!08\)\( T + \)\(65\!\cdots\!88\)\( p T^{2} - \)\(38\!\cdots\!72\)\( p^{3} T^{3} + \)\(12\!\cdots\!50\)\( p^{5} T^{4} - \)\(38\!\cdots\!72\)\( p^{44} T^{5} + \)\(65\!\cdots\!88\)\( p^{83} T^{6} + \)\(88\!\cdots\!08\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
17 | $C_2 \wr S_4$ | \( 1 - \)\(38\!\cdots\!88\)\( T + \)\(68\!\cdots\!16\)\( p T^{2} - \)\(48\!\cdots\!60\)\( p^{3} T^{3} + \)\(33\!\cdots\!18\)\( p^{5} T^{4} - \)\(48\!\cdots\!60\)\( p^{44} T^{5} + \)\(68\!\cdots\!16\)\( p^{83} T^{6} - \)\(38\!\cdots\!88\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
19 | $C_2 \wr S_4$ | \( 1 - \)\(26\!\cdots\!48\)\( T + \)\(39\!\cdots\!48\)\( p T^{2} - \)\(21\!\cdots\!64\)\( p^{3} T^{3} + \)\(10\!\cdots\!66\)\( p^{5} T^{4} - \)\(21\!\cdots\!64\)\( p^{44} T^{5} + \)\(39\!\cdots\!48\)\( p^{83} T^{6} - \)\(26\!\cdots\!48\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
23 | $C_2 \wr S_4$ | \( 1 - \)\(15\!\cdots\!32\)\( T + \)\(86\!\cdots\!76\)\( p T^{2} - \)\(28\!\cdots\!52\)\( p^{2} T^{3} + \)\(10\!\cdots\!50\)\( p^{3} T^{4} - \)\(28\!\cdots\!52\)\( p^{43} T^{5} + \)\(86\!\cdots\!76\)\( p^{83} T^{6} - \)\(15\!\cdots\!32\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
29 | $C_2 \wr S_4$ | \( 1 - \)\(10\!\cdots\!64\)\( T + \)\(94\!\cdots\!40\)\( p T^{2} - \)\(31\!\cdots\!88\)\( p^{2} T^{3} + \)\(13\!\cdots\!82\)\( p^{3} T^{4} - \)\(31\!\cdots\!88\)\( p^{43} T^{5} + \)\(94\!\cdots\!40\)\( p^{83} T^{6} - \)\(10\!\cdots\!64\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
31 | $C_2 \wr S_4$ | \( 1 - \)\(92\!\cdots\!04\)\( T + \)\(25\!\cdots\!88\)\( p T^{2} - \)\(41\!\cdots\!52\)\( p^{2} T^{3} + \)\(59\!\cdots\!54\)\( p^{3} T^{4} - \)\(41\!\cdots\!52\)\( p^{43} T^{5} + \)\(25\!\cdots\!88\)\( p^{83} T^{6} - \)\(92\!\cdots\!04\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
37 | $C_2 \wr S_4$ | \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(14\!\cdots\!16\)\( p T^{2} - \)\(62\!\cdots\!84\)\( p^{2} T^{3} + \)\(31\!\cdots\!90\)\( p^{3} T^{4} - \)\(62\!\cdots\!84\)\( p^{43} T^{5} + \)\(14\!\cdots\!16\)\( p^{83} T^{6} - \)\(20\!\cdots\!56\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
41 | $C_2 \wr S_4$ | \( 1 + \)\(23\!\cdots\!04\)\( T + \)\(12\!\cdots\!88\)\( p T^{2} + \)\(70\!\cdots\!72\)\( T^{3} + \)\(99\!\cdots\!94\)\( T^{4} + \)\(70\!\cdots\!72\)\( p^{41} T^{5} + \)\(12\!\cdots\!88\)\( p^{83} T^{6} + \)\(23\!\cdots\!04\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
43 | $C_2 \wr S_4$ | \( 1 - \)\(39\!\cdots\!60\)\( T + \)\(29\!\cdots\!00\)\( T^{2} - \)\(85\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!98\)\( T^{4} - \)\(85\!\cdots\!80\)\( p^{41} T^{5} + \)\(29\!\cdots\!00\)\( p^{82} T^{6} - \)\(39\!\cdots\!60\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
47 | $C_2 \wr S_4$ | \( 1 - \)\(88\!\cdots\!20\)\( T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!18\)\( T^{4} - \)\(42\!\cdots\!40\)\( p^{41} T^{5} + \)\(10\!\cdots\!20\)\( p^{82} T^{6} - \)\(88\!\cdots\!20\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
53 | $C_2 \wr S_4$ | \( 1 + \)\(95\!\cdots\!28\)\( T + \)\(13\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(85\!\cdots\!10\)\( T^{4} + \)\(10\!\cdots\!72\)\( p^{41} T^{5} + \)\(13\!\cdots\!88\)\( p^{82} T^{6} + \)\(95\!\cdots\!28\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
59 | $C_2 \wr S_4$ | \( 1 - \)\(18\!\cdots\!08\)\( T + \)\(66\!\cdots\!52\)\( T^{2} - \)\(24\!\cdots\!16\)\( T^{3} + \)\(18\!\cdots\!74\)\( T^{4} - \)\(24\!\cdots\!16\)\( p^{41} T^{5} + \)\(66\!\cdots\!52\)\( p^{82} T^{6} - \)\(18\!\cdots\!08\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
61 | $C_2 \wr S_4$ | \( 1 - \)\(53\!\cdots\!40\)\( T + \)\(67\!\cdots\!56\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!26\)\( T^{4} - \)\(24\!\cdots\!60\)\( p^{41} T^{5} + \)\(67\!\cdots\!56\)\( p^{82} T^{6} - \)\(53\!\cdots\!40\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
67 | $C_2 \wr S_4$ | \( 1 + \)\(73\!\cdots\!28\)\( T + \)\(45\!\cdots\!12\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!06\)\( T^{4} + \)\(17\!\cdots\!00\)\( p^{41} T^{5} + \)\(45\!\cdots\!12\)\( p^{82} T^{6} + \)\(73\!\cdots\!28\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
71 | $C_2 \wr S_4$ | \( 1 - \)\(84\!\cdots\!52\)\( T + \)\(22\!\cdots\!48\)\( T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} - \)\(13\!\cdots\!64\)\( p^{41} T^{5} + \)\(22\!\cdots\!48\)\( p^{82} T^{6} - \)\(84\!\cdots\!52\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
73 | $C_2 \wr S_4$ | \( 1 + \)\(44\!\cdots\!32\)\( T + \)\(51\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} - \)\(16\!\cdots\!72\)\( p^{41} T^{5} + \)\(51\!\cdots\!68\)\( p^{82} T^{6} + \)\(44\!\cdots\!32\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
79 | $C_2 \wr S_4$ | \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(22\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!46\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{41} T^{5} + \)\(22\!\cdots\!16\)\( p^{82} T^{6} - \)\(14\!\cdots\!80\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
83 | $C_2 \wr S_4$ | \( 1 + \)\(15\!\cdots\!04\)\( T + \)\(99\!\cdots\!00\)\( T^{2} + \)\(72\!\cdots\!24\)\( T^{3} + \)\(39\!\cdots\!46\)\( T^{4} + \)\(72\!\cdots\!24\)\( p^{41} T^{5} + \)\(99\!\cdots\!00\)\( p^{82} T^{6} + \)\(15\!\cdots\!04\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
89 | $C_2 \wr S_4$ | \( 1 - \)\(39\!\cdots\!72\)\( T + \)\(90\!\cdots\!12\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!46\)\( p T^{4} - \)\(13\!\cdots\!84\)\( p^{41} T^{5} + \)\(90\!\cdots\!12\)\( p^{82} T^{6} - \)\(39\!\cdots\!72\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
97 | $C_2 \wr S_4$ | \( 1 - \)\(36\!\cdots\!52\)\( T + \)\(23\!\cdots\!52\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} - \)\(59\!\cdots\!94\)\( T^{4} + \)\(31\!\cdots\!80\)\( p^{41} T^{5} + \)\(23\!\cdots\!52\)\( p^{82} T^{6} - \)\(36\!\cdots\!52\)\( p^{123} T^{7} + p^{164} T^{8} \) | |
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Imaginary part of the first few zeros on the critical line
−8.331020463592550236672127453641, −7.79993263797261049534871113712, −7.69547013582766612117479682604, −7.68797501040805993089930870565, −6.99702201552922234511984862425, −6.72494904175974151075211110538, −6.17035242993108852474155209268, −5.98168465454814581480539571789, −5.67598560151129529596397977482, −4.95382008337360170172465857267, −4.86910847249766959110978771037, −4.78392744388453468911577115156, −4.64159519385288998489741452472, −3.78619763391205899425437370767, −3.44308893940433742433320920963, −3.39419904258618659994002799967, −3.06988901815478806955305517106, −2.50869485758533505581275231364, −2.50572414961354334498396170177, −1.78243812513803066711565897812, −1.45871595840921031519091056326, −1.04194298822418418545648886719, −0.65624621430297348650834640855, −0.58942730796729157139672458321, −0.54275648870626797426047474666, 0.54275648870626797426047474666, 0.58942730796729157139672458321, 0.65624621430297348650834640855, 1.04194298822418418545648886719, 1.45871595840921031519091056326, 1.78243812513803066711565897812, 2.50572414961354334498396170177, 2.50869485758533505581275231364, 3.06988901815478806955305517106, 3.39419904258618659994002799967, 3.44308893940433742433320920963, 3.78619763391205899425437370767, 4.64159519385288998489741452472, 4.78392744388453468911577115156, 4.86910847249766959110978771037, 4.95382008337360170172465857267, 5.67598560151129529596397977482, 5.98168465454814581480539571789, 6.17035242993108852474155209268, 6.72494904175974151075211110538, 6.99702201552922234511984862425, 7.68797501040805993089930870565, 7.69547013582766612117479682604, 7.79993263797261049534871113712, 8.331020463592550236672127453641