Dirichlet series
| L(s) = 1 | − 2.17e8·2-s + 3.74e13·3-s − 1.17e17·4-s − 1.06e20·5-s − 8.16e21·6-s + 9.52e23·7-s + 1.00e25·8-s − 2.03e27·9-s + 2.32e28·10-s − 4.62e29·11-s − 4.42e30·12-s − 3.56e31·13-s − 2.07e32·14-s − 4.00e33·15-s + 3.75e33·16-s + 1.38e34·17-s + 4.42e35·18-s + 3.36e35·19-s + 1.26e37·20-s + 3.57e37·21-s + 1.00e38·22-s − 2.97e39·23-s + 3.77e38·24-s − 1.28e40·25-s + 7.76e39·26-s − 8.79e40·27-s − 1.12e41·28-s + ⋯ |
| L(s) = 1 | − 0.573·2-s + 0.945·3-s − 0.818·4-s − 1.28·5-s − 0.542·6-s + 0.782·7-s + 0.184·8-s − 1.29·9-s + 0.736·10-s − 0.966·11-s − 0.774·12-s − 0.638·13-s − 0.449·14-s − 1.21·15-s + 0.180·16-s + 0.118·17-s + 0.743·18-s + 0.121·19-s + 1.05·20-s + 0.740·21-s + 0.554·22-s − 4.61·23-s + 0.174·24-s − 1.85·25-s + 0.366·26-s − 1.41·27-s − 0.640·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(58-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(179266.\) |
| Root analytic conductor: | \(4.53614\) |
| Motivic weight: | \(57\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 1,\ (\ :57/2, 57/2, 57/2, 57/2),\ 1)\) |
Particular Values
| \(L(29)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{59}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| good | 2 | $C_2 \wr S_4$ | \( 1 + 13609035 p^{4} T + 10093644696095 p^{14} T^{2} + 769272252793992405 p^{26} T^{3} + \)\(22\!\cdots\!93\)\( p^{40} T^{4} + 769272252793992405 p^{83} T^{5} + 10093644696095 p^{128} T^{6} + 13609035 p^{175} T^{7} + p^{228} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - 1387994895280 p^{3} T + \)\(17\!\cdots\!20\)\( p^{9} T^{2} - \)\(30\!\cdots\!20\)\( p^{18} T^{3} + \)\(12\!\cdots\!86\)\( p^{29} T^{4} - \)\(30\!\cdots\!20\)\( p^{75} T^{5} + \)\(17\!\cdots\!20\)\( p^{123} T^{6} - 1387994895280 p^{174} T^{7} + p^{228} T^{8} \) | |
| 5 | $C_2 \wr S_4$ | \( 1 + 4278451048313707944 p^{2} T + \)\(31\!\cdots\!96\)\( p^{7} T^{2} + \)\(32\!\cdots\!24\)\( p^{14} T^{3} + \)\(10\!\cdots\!14\)\( p^{22} T^{4} + \)\(32\!\cdots\!24\)\( p^{71} T^{5} + \)\(31\!\cdots\!96\)\( p^{121} T^{6} + 4278451048313707944 p^{173} T^{7} + p^{228} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 - \)\(13\!\cdots\!00\)\( p T + \)\(16\!\cdots\!00\)\( p^{5} T^{2} - \)\(21\!\cdots\!00\)\( p^{10} T^{3} + \)\(89\!\cdots\!98\)\( p^{16} T^{4} - \)\(21\!\cdots\!00\)\( p^{67} T^{5} + \)\(16\!\cdots\!00\)\( p^{119} T^{6} - \)\(13\!\cdots\!00\)\( p^{172} T^{7} + p^{228} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + \)\(42\!\cdots\!92\)\( p T + \)\(28\!\cdots\!48\)\( p^{3} T^{2} + \)\(29\!\cdots\!24\)\( p^{6} T^{3} + \)\(24\!\cdots\!70\)\( p^{9} T^{4} + \)\(29\!\cdots\!24\)\( p^{63} T^{5} + \)\(28\!\cdots\!48\)\( p^{117} T^{6} + \)\(42\!\cdots\!92\)\( p^{172} T^{7} + p^{228} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + \)\(27\!\cdots\!60\)\( p T + \)\(38\!\cdots\!60\)\( p^{3} T^{2} + \)\(48\!\cdots\!60\)\( p^{6} T^{3} + \)\(23\!\cdots\!22\)\( p^{10} T^{4} + \)\(48\!\cdots\!60\)\( p^{63} T^{5} + \)\(38\!\cdots\!60\)\( p^{117} T^{6} + \)\(27\!\cdots\!60\)\( p^{172} T^{7} + p^{228} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - \)\(81\!\cdots\!60\)\( p T + \)\(62\!\cdots\!80\)\( p^{3} T^{2} - \)\(89\!\cdots\!20\)\( p^{5} T^{3} + \)\(79\!\cdots\!38\)\( p^{8} T^{4} - \)\(89\!\cdots\!20\)\( p^{62} T^{5} + \)\(62\!\cdots\!80\)\( p^{117} T^{6} - \)\(81\!\cdots\!60\)\( p^{172} T^{7} + p^{228} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - \)\(17\!\cdots\!40\)\( p T + \)\(35\!\cdots\!96\)\( p^{2} T^{2} + \)\(32\!\cdots\!20\)\( p^{3} T^{3} + \)\(61\!\cdots\!06\)\( p^{4} T^{4} + \)\(32\!\cdots\!20\)\( p^{60} T^{5} + \)\(35\!\cdots\!96\)\( p^{116} T^{6} - \)\(17\!\cdots\!40\)\( p^{172} T^{7} + p^{228} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + \)\(12\!\cdots\!40\)\( p T + \)\(89\!\cdots\!20\)\( p^{2} T^{2} + \)\(40\!\cdots\!80\)\( p^{3} T^{3} + \)\(58\!\cdots\!26\)\( p^{5} T^{4} + \)\(40\!\cdots\!80\)\( p^{60} T^{5} + \)\(89\!\cdots\!20\)\( p^{116} T^{6} + \)\(12\!\cdots\!40\)\( p^{172} T^{7} + p^{228} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - \)\(69\!\cdots\!40\)\( T + \)\(46\!\cdots\!36\)\( T^{2} - \)\(36\!\cdots\!20\)\( p T^{3} + \)\(66\!\cdots\!46\)\( p^{2} T^{4} - \)\(36\!\cdots\!20\)\( p^{58} T^{5} + \)\(46\!\cdots\!36\)\( p^{114} T^{6} - \)\(69\!\cdots\!40\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + \)\(89\!\cdots\!52\)\( T + \)\(23\!\cdots\!08\)\( T^{2} - \)\(19\!\cdots\!16\)\( p T^{3} + \)\(26\!\cdots\!70\)\( p^{2} T^{4} - \)\(19\!\cdots\!16\)\( p^{58} T^{5} + \)\(23\!\cdots\!08\)\( p^{114} T^{6} + \)\(89\!\cdots\!52\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(35\!\cdots\!60\)\( p T^{2} - \)\(58\!\cdots\!80\)\( p^{2} T^{3} + \)\(10\!\cdots\!26\)\( p^{3} T^{4} - \)\(58\!\cdots\!80\)\( p^{59} T^{5} + \)\(35\!\cdots\!60\)\( p^{115} T^{6} - \)\(10\!\cdots\!60\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + \)\(11\!\cdots\!72\)\( T + \)\(85\!\cdots\!48\)\( p T^{2} + \)\(16\!\cdots\!04\)\( p^{2} T^{3} + \)\(64\!\cdots\!70\)\( p^{3} T^{4} + \)\(16\!\cdots\!04\)\( p^{59} T^{5} + \)\(85\!\cdots\!48\)\( p^{115} T^{6} + \)\(11\!\cdots\!72\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + \)\(61\!\cdots\!00\)\( T + \)\(36\!\cdots\!00\)\( p T^{2} - \)\(47\!\cdots\!00\)\( p^{2} T^{3} - \)\(65\!\cdots\!86\)\( p^{3} T^{4} - \)\(47\!\cdots\!00\)\( p^{59} T^{5} + \)\(36\!\cdots\!00\)\( p^{115} T^{6} + \)\(61\!\cdots\!00\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - \)\(36\!\cdots\!80\)\( T + \)\(82\!\cdots\!80\)\( p T^{2} + \)\(14\!\cdots\!60\)\( p^{2} T^{3} + \)\(37\!\cdots\!06\)\( p^{3} T^{4} + \)\(14\!\cdots\!60\)\( p^{59} T^{5} + \)\(82\!\cdots\!80\)\( p^{115} T^{6} - \)\(36\!\cdots\!80\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + \)\(66\!\cdots\!80\)\( p T + \)\(32\!\cdots\!40\)\( p^{2} T^{2} + \)\(10\!\cdots\!60\)\( p^{3} T^{3} + \)\(31\!\cdots\!98\)\( p^{4} T^{4} + \)\(10\!\cdots\!60\)\( p^{60} T^{5} + \)\(32\!\cdots\!40\)\( p^{116} T^{6} + \)\(66\!\cdots\!80\)\( p^{172} T^{7} + p^{228} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - \)\(67\!\cdots\!80\)\( T + \)\(32\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!66\)\( T^{4} - \)\(17\!\cdots\!60\)\( p^{57} T^{5} + \)\(32\!\cdots\!76\)\( p^{114} T^{6} - \)\(67\!\cdots\!80\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + \)\(15\!\cdots\!12\)\( T + \)\(51\!\cdots\!08\)\( p T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(29\!\cdots\!70\)\( T^{4} + \)\(28\!\cdots\!64\)\( p^{57} T^{5} + \)\(51\!\cdots\!08\)\( p^{115} T^{6} + \)\(15\!\cdots\!12\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - \)\(15\!\cdots\!20\)\( T + \)\(36\!\cdots\!40\)\( T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + \)\(64\!\cdots\!58\)\( T^{4} - \)\(45\!\cdots\!40\)\( p^{57} T^{5} + \)\(36\!\cdots\!40\)\( p^{114} T^{6} - \)\(15\!\cdots\!20\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + \)\(73\!\cdots\!32\)\( T + \)\(12\!\cdots\!48\)\( T^{2} + \)\(61\!\cdots\!84\)\( T^{3} + \)\(59\!\cdots\!70\)\( T^{4} + \)\(61\!\cdots\!84\)\( p^{57} T^{5} + \)\(12\!\cdots\!48\)\( p^{114} T^{6} + \)\(73\!\cdots\!32\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(59\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!18\)\( T^{4} + \)\(56\!\cdots\!60\)\( p^{57} T^{5} + \)\(59\!\cdots\!80\)\( p^{114} T^{6} + \)\(13\!\cdots\!20\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + \)\(82\!\cdots\!60\)\( T + \)\(56\!\cdots\!36\)\( T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(34\!\cdots\!20\)\( p^{57} T^{5} + \)\(56\!\cdots\!36\)\( p^{114} T^{6} + \)\(82\!\cdots\!60\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + \)\(98\!\cdots\!60\)\( T + \)\(12\!\cdots\!40\)\( T^{2} + \)\(72\!\cdots\!80\)\( T^{3} + \)\(48\!\cdots\!58\)\( T^{4} + \)\(72\!\cdots\!80\)\( p^{57} T^{5} + \)\(12\!\cdots\!40\)\( p^{114} T^{6} + \)\(98\!\cdots\!60\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(35\!\cdots\!16\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!46\)\( T^{4} + \)\(17\!\cdots\!60\)\( p^{57} T^{5} + \)\(35\!\cdots\!16\)\( p^{114} T^{6} + \)\(10\!\cdots\!80\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + \)\(15\!\cdots\!20\)\( T + \)\(10\!\cdots\!60\)\( T^{2} + \)\(68\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!38\)\( T^{4} + \)\(68\!\cdots\!40\)\( p^{57} T^{5} + \)\(10\!\cdots\!60\)\( p^{114} T^{6} + \)\(15\!\cdots\!20\)\( p^{171} T^{7} + p^{228} T^{8} \) | |
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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
−13.77887444428464029284883017171, −13.06526190531737447374786266583, −12.20477665507040089983111728082, −11.88546095742190219724901706354, −11.60913412900916759381674449433, −11.15650724356567680291578643487, −10.14981542898063551358400840658, −9.781985696962920311432057690200, −9.655328233176605859002595396270, −8.539143718486060108935929449987, −8.463727629061523807859670302970, −8.084656797984681345198693411590, −7.72492792130318511772563790365, −7.67748387239562890329726998409, −6.22178203762451426384050184986, −6.04040700297207970322893165931, −5.44652409327183811555795234813, −4.63143830175394192625812911713, −4.40982350389365240746261505984, −3.82904420000498583906991342567, −3.43596312250581772988677705601, −2.61705916330572294438575489244, −2.60299345341729400408032905577, −1.76084178090973606619894531008, −1.44458006649267949059849455512, 0, 0, 0, 0, 1.44458006649267949059849455512, 1.76084178090973606619894531008, 2.60299345341729400408032905577, 2.61705916330572294438575489244, 3.43596312250581772988677705601, 3.82904420000498583906991342567, 4.40982350389365240746261505984, 4.63143830175394192625812911713, 5.44652409327183811555795234813, 6.04040700297207970322893165931, 6.22178203762451426384050184986, 7.67748387239562890329726998409, 7.72492792130318511772563790365, 8.084656797984681345198693411590, 8.463727629061523807859670302970, 8.539143718486060108935929449987, 9.655328233176605859002595396270, 9.781985696962920311432057690200, 10.14981542898063551358400840658, 11.15650724356567680291578643487, 11.60913412900916759381674449433, 11.88546095742190219724901706354, 12.20477665507040089983111728082, 13.06526190531737447374786266583, 13.77887444428464029284883017171