Dirichlet series
| L(s) = 1 | − 128·2-s − 972·3-s + 1.02e4·4-s − 9.60e3·5-s + 1.24e5·6-s − 1.39e4·7-s − 6.55e5·8-s + 5.90e5·9-s + 1.22e6·10-s + 1.40e4·11-s − 9.95e6·12-s − 1.51e5·13-s + 1.78e6·14-s + 9.33e6·15-s + 3.67e7·16-s + 1.87e6·17-s − 7.55e7·18-s − 9.90e6·19-s − 9.83e7·20-s + 1.35e7·21-s − 1.79e6·22-s + 3.73e7·23-s + 6.37e8·24-s − 4.59e7·25-s + 1.93e7·26-s − 2.86e8·27-s − 1.42e8·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s − 1.37·5-s + 6.53·6-s − 0.313·7-s − 7.07·8-s + 10/3·9-s + 3.88·10-s + 0.0263·11-s − 11.5·12-s − 0.113·13-s + 0.885·14-s + 3.17·15-s + 35/4·16-s + 0.319·17-s − 9.42·18-s − 0.917·19-s − 6.87·20-s + 0.723·21-s − 0.0744·22-s + 1.21·23-s + 16.3·24-s − 0.940·25-s + 0.319·26-s − 3.84·27-s − 1.56·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.88627\times 10^{7}\) |
| Root analytic conductor: | \(9.35901\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(0.0004169626857\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0004169626857\) |
| \(L(\frac{13}{2})\) | 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 | 2 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 9604 T + 27632857 p T^{2} + 267565646 p^{5} T^{3} + 69343879448552 p^{3} T^{4} + 267565646 p^{16} T^{5} + 27632857 p^{23} T^{6} + 9604 p^{33} T^{7} + p^{44} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 13926 T + 5481150313 T^{2} - 152507798874 p T^{3} + 267490363253985468 p^{2} T^{4} - 152507798874 p^{12} T^{5} + 5481150313 p^{22} T^{6} + 13926 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 14060 T + 39042590227 p T^{2} + 255002398063781062 T^{3} + \)\(56\!\cdots\!40\)\( T^{4} + 255002398063781062 p^{11} T^{5} + 39042590227 p^{23} T^{6} - 14060 p^{33} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 151318 T + 2544781392832 T^{2} - 1778631945735484718 T^{3} + \)\(26\!\cdots\!70\)\( T^{4} - 1778631945735484718 p^{11} T^{5} + 2544781392832 p^{22} T^{6} + 151318 p^{33} T^{7} + p^{44} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 1872174 T + 47098907983157 T^{2} + 57952352166511137210 T^{3} + \)\(81\!\cdots\!20\)\( T^{4} + 57952352166511137210 p^{11} T^{5} + 47098907983157 p^{22} T^{6} - 1872174 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 37375844 T + 2199106192902896 T^{2} - \)\(60\!\cdots\!28\)\( T^{3} + \)\(29\!\cdots\!38\)\( T^{4} - \)\(60\!\cdots\!28\)\( p^{11} T^{5} + 2199106192902896 p^{22} T^{6} - 37375844 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 193432382 T + 31235055978927848 T^{2} + \)\(40\!\cdots\!58\)\( T^{3} + \)\(54\!\cdots\!22\)\( T^{4} + \)\(40\!\cdots\!58\)\( p^{11} T^{5} + 31235055978927848 p^{22} T^{6} + 193432382 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 129188026 T + 68279162251391320 T^{2} - \)\(86\!\cdots\!02\)\( T^{3} + \)\(22\!\cdots\!94\)\( T^{4} - \)\(86\!\cdots\!02\)\( p^{11} T^{5} + 68279162251391320 p^{22} T^{6} - 129188026 p^{33} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 216638518 T + 354652071392699188 T^{2} - \)\(79\!\cdots\!14\)\( T^{3} + \)\(93\!\cdots\!18\)\( T^{4} - \)\(79\!\cdots\!14\)\( p^{11} T^{5} + 354652071392699188 p^{22} T^{6} - 216638518 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 1230483434 T + 2403447383841007748 T^{2} + \)\(19\!\cdots\!22\)\( T^{3} + \)\(20\!\cdots\!30\)\( T^{4} + \)\(19\!\cdots\!22\)\( p^{11} T^{5} + 2403447383841007748 p^{22} T^{6} + 1230483434 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 2313960998 T + 4466992207187618617 T^{2} + \)\(62\!\cdots\!26\)\( T^{3} + \)\(65\!\cdots\!08\)\( T^{4} + \)\(62\!\cdots\!26\)\( p^{11} T^{5} + 4466992207187618617 p^{22} T^{6} + 2313960998 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 5973606190 T + 22333418067986635433 T^{2} + \)\(54\!\cdots\!30\)\( T^{3} + \)\(10\!\cdots\!28\)\( T^{4} + \)\(54\!\cdots\!30\)\( p^{11} T^{5} + 22333418067986635433 p^{22} T^{6} + 5973606190 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 8305233754 T + 52082942403716556680 T^{2} + \)\(20\!\cdots\!26\)\( T^{3} + \)\(71\!\cdots\!74\)\( T^{4} + \)\(20\!\cdots\!26\)\( p^{11} T^{5} + 52082942403716556680 p^{22} T^{6} + 8305233754 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 4934963400 T + 52448512906692256892 T^{2} + \)\(26\!\cdots\!12\)\( T^{3} + \)\(26\!\cdots\!34\)\( T^{4} + \)\(26\!\cdots\!12\)\( p^{11} T^{5} + 52448512906692256892 p^{22} T^{6} + 4934963400 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 9210503722 T + 29916278542163139913 T^{2} - \)\(14\!\cdots\!46\)\( T^{3} - \)\(18\!\cdots\!80\)\( T^{4} - \)\(14\!\cdots\!46\)\( p^{11} T^{5} + 29916278542163139913 p^{22} T^{6} + 9210503722 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 4161437760 T + \)\(46\!\cdots\!68\)\( T^{2} + \)\(15\!\cdots\!56\)\( T^{3} + \)\(84\!\cdots\!74\)\( T^{4} + \)\(15\!\cdots\!56\)\( p^{11} T^{5} + \)\(46\!\cdots\!68\)\( p^{22} T^{6} + 4161437760 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 5071750024 T + \)\(36\!\cdots\!84\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(90\!\cdots\!66\)\( T^{4} + \)\(40\!\cdots\!48\)\( p^{11} T^{5} + \)\(36\!\cdots\!84\)\( p^{22} T^{6} + 5071750024 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1430676858 T + \)\(65\!\cdots\!09\)\( T^{2} + \)\(21\!\cdots\!18\)\( T^{3} + \)\(27\!\cdots\!12\)\( T^{4} + \)\(21\!\cdots\!18\)\( p^{11} T^{5} + \)\(65\!\cdots\!09\)\( p^{22} T^{6} + 1430676858 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 30371714270 T + \)\(30\!\cdots\!16\)\( T^{2} - \)\(65\!\cdots\!98\)\( T^{3} + \)\(34\!\cdots\!46\)\( T^{4} - \)\(65\!\cdots\!98\)\( p^{11} T^{5} + \)\(30\!\cdots\!16\)\( p^{22} T^{6} - 30371714270 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 6157448418 T + \)\(47\!\cdots\!88\)\( T^{2} + \)\(22\!\cdots\!66\)\( T^{3} + \)\(88\!\cdots\!98\)\( T^{4} + \)\(22\!\cdots\!66\)\( p^{11} T^{5} + \)\(47\!\cdots\!88\)\( p^{22} T^{6} + 6157448418 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 106935172590 T + \)\(85\!\cdots\!32\)\( T^{2} - \)\(30\!\cdots\!38\)\( T^{3} + \)\(16\!\cdots\!34\)\( T^{4} - \)\(30\!\cdots\!38\)\( p^{11} T^{5} + \)\(85\!\cdots\!32\)\( p^{22} T^{6} - 106935172590 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 222927041956 T + \)\(43\!\cdots\!80\)\( T^{2} - \)\(51\!\cdots\!36\)\( T^{3} + \)\(52\!\cdots\!30\)\( T^{4} - \)\(51\!\cdots\!36\)\( p^{11} T^{5} + \)\(43\!\cdots\!80\)\( p^{22} T^{6} - 222927041956 p^{33} T^{7} + p^{44} T^{8} \) | |
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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
−7.964832704761174558750956425332, −7.39940228850175260009902356050, −7.39625423615316707696512367972, −7.09771038470808227844933605992, −6.76011500687699615187397779970, −6.29660010175357224950112206979, −6.28789398724740600261962500678, −6.07273197325851943516240353718, −5.91414492688965491380307134533, −5.00576942807452840557203451199, −4.95703185012693861482553396839, −4.64474678999415570735327307847, −4.60119527575772703421978135964, −3.54045840948444008322237473402, −3.46375461923850629395549239146, −3.18592894009149960929665061480, −3.16342421088442609509064888685, −1.94426683944389346218830629268, −1.82645193415456712747656906305, −1.73150834098429238534286874947, −1.58100823883263820436545768666, −0.76625518173310596921072936116, −0.68620165131838237361039969977, −0.26840729271803415916436264153, −0.01567173769009754121593357013, 0.01567173769009754121593357013, 0.26840729271803415916436264153, 0.68620165131838237361039969977, 0.76625518173310596921072936116, 1.58100823883263820436545768666, 1.73150834098429238534286874947, 1.82645193415456712747656906305, 1.94426683944389346218830629268, 3.16342421088442609509064888685, 3.18592894009149960929665061480, 3.46375461923850629395549239146, 3.54045840948444008322237473402, 4.60119527575772703421978135964, 4.64474678999415570735327307847, 4.95703185012693861482553396839, 5.00576942807452840557203451199, 5.91414492688965491380307134533, 6.07273197325851943516240353718, 6.28789398724740600261962500678, 6.29660010175357224950112206979, 6.76011500687699615187397779970, 7.09771038470808227844933605992, 7.39625423615316707696512367972, 7.39940228850175260009902356050, 7.964832704761174558750956425332