Properties

Degree $8$
Conductor $1$
Sign $1$
Motivic weight $61$
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.14e9·2-s − 5.73e14·3-s − 1.75e18·4-s − 5.24e20·5-s − 6.57e23·6-s − 6.33e25·7-s − 1.36e26·8-s − 3.14e28·9-s − 6.00e29·10-s − 4.15e31·11-s + 1.00e33·12-s + 1.07e34·13-s − 7.26e34·14-s + 3.00e35·15-s + 4.30e36·16-s − 4.09e37·17-s − 3.60e37·18-s − 3.56e38·19-s + 9.18e38·20-s + 3.63e40·21-s − 4.76e40·22-s − 4.10e41·23-s + 7.85e40·24-s − 1.69e43·25-s + 1.23e43·26-s + 6.25e43·27-s + 1.11e44·28-s + ⋯
L(s)  = 1  + 0.754·2-s − 1.60·3-s − 0.760·4-s − 0.251·5-s − 1.21·6-s − 1.06·7-s − 0.0390·8-s − 0.247·9-s − 0.189·10-s − 0.718·11-s + 1.22·12-s + 1.13·13-s − 0.801·14-s + 0.404·15-s + 0.809·16-s − 1.21·17-s − 0.186·18-s − 0.354·19-s + 0.191·20-s + 1.70·21-s − 0.542·22-s − 1.20·23-s + 0.0628·24-s − 3.89·25-s + 0.860·26-s + 1.37·27-s + 0.807·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(62-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1\)
Sign: $1$
Motivic weight: \(61\)
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 1,\ (\ :61/2, 61/2, 61/2, 61/2),\ 1)\)

Particular Values

\(L(31)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{63}{2})\) 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)$
good2$C_2 \wr S_4$ \( 1 - 17911125 p^{6} T + 748828804000265 p^{12} T^{2} - 80298767510162670375 p^{26} T^{3} + \)\(64\!\cdots\!33\)\( p^{40} T^{4} - 80298767510162670375 p^{87} T^{5} + 748828804000265 p^{134} T^{6} - 17911125 p^{189} T^{7} + p^{244} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 21249022186000 p^{3} T + \)\(61\!\cdots\!20\)\( p^{10} T^{2} + \)\(13\!\cdots\!00\)\( p^{19} T^{3} + \)\(11\!\cdots\!94\)\( p^{31} T^{4} + \)\(13\!\cdots\!00\)\( p^{80} T^{5} + \)\(61\!\cdots\!20\)\( p^{132} T^{6} + 21249022186000 p^{186} T^{7} + p^{244} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 20965039840811809512 p^{2} T + \)\(21\!\cdots\!72\)\( p^{7} T^{2} + \)\(55\!\cdots\!92\)\( p^{13} T^{3} + \)\(46\!\cdots\!26\)\( p^{22} T^{4} + \)\(55\!\cdots\!92\)\( p^{74} T^{5} + \)\(21\!\cdots\!72\)\( p^{129} T^{6} + 20965039840811809512 p^{185} T^{7} + p^{244} T^{8} \)
7$C_2 \wr S_4$ \( 1 + \)\(12\!\cdots\!00\)\( p^{2} T + \)\(71\!\cdots\!00\)\( p^{6} T^{2} + \)\(24\!\cdots\!00\)\( p^{11} T^{3} + \)\(23\!\cdots\!02\)\( p^{18} T^{4} + \)\(24\!\cdots\!00\)\( p^{72} T^{5} + \)\(71\!\cdots\!00\)\( p^{128} T^{6} + \)\(12\!\cdots\!00\)\( p^{185} T^{7} + p^{244} T^{8} \)
11$C_2 \wr S_4$ \( 1 + \)\(41\!\cdots\!52\)\( T + \)\(57\!\cdots\!48\)\( p^{2} T^{2} + \)\(15\!\cdots\!04\)\( p^{5} T^{3} + \)\(10\!\cdots\!70\)\( p^{9} T^{4} + \)\(15\!\cdots\!04\)\( p^{66} T^{5} + \)\(57\!\cdots\!48\)\( p^{124} T^{6} + \)\(41\!\cdots\!52\)\( p^{183} T^{7} + p^{244} T^{8} \)
13$C_2 \wr S_4$ \( 1 - \)\(82\!\cdots\!00\)\( p T + \)\(79\!\cdots\!80\)\( p^{3} T^{2} - \)\(25\!\cdots\!00\)\( p^{6} T^{3} + \)\(10\!\cdots\!62\)\( p^{10} T^{4} - \)\(25\!\cdots\!00\)\( p^{67} T^{5} + \)\(79\!\cdots\!80\)\( p^{125} T^{6} - \)\(82\!\cdots\!00\)\( p^{184} T^{7} + p^{244} T^{8} \)
17$C_2 \wr S_4$ \( 1 + \)\(24\!\cdots\!00\)\( p T + \)\(14\!\cdots\!80\)\( p^{2} T^{2} + \)\(12\!\cdots\!00\)\( p^{4} T^{3} + \)\(15\!\cdots\!86\)\( p^{7} T^{4} + \)\(12\!\cdots\!00\)\( p^{65} T^{5} + \)\(14\!\cdots\!80\)\( p^{124} T^{6} + \)\(24\!\cdots\!00\)\( p^{184} T^{7} + p^{244} T^{8} \)
19$C_2 \wr S_4$ \( 1 + \)\(18\!\cdots\!80\)\( p T + \)\(23\!\cdots\!64\)\( p^{3} T^{2} + \)\(11\!\cdots\!60\)\( p^{5} T^{3} + \)\(11\!\cdots\!26\)\( p^{8} T^{4} + \)\(11\!\cdots\!60\)\( p^{66} T^{5} + \)\(23\!\cdots\!64\)\( p^{125} T^{6} + \)\(18\!\cdots\!80\)\( p^{184} T^{7} + p^{244} T^{8} \)
23$C_2 \wr S_4$ \( 1 + \)\(41\!\cdots\!00\)\( T + \)\(18\!\cdots\!80\)\( p T^{2} + \)\(25\!\cdots\!00\)\( p^{2} T^{3} + \)\(26\!\cdots\!38\)\( p^{4} T^{4} + \)\(25\!\cdots\!00\)\( p^{63} T^{5} + \)\(18\!\cdots\!80\)\( p^{123} T^{6} + \)\(41\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
29$C_2 \wr S_4$ \( 1 - \)\(23\!\cdots\!80\)\( p T + \)\(51\!\cdots\!76\)\( p^{2} T^{2} + \)\(65\!\cdots\!60\)\( p^{4} T^{3} + \)\(14\!\cdots\!26\)\( p^{6} T^{4} + \)\(65\!\cdots\!60\)\( p^{65} T^{5} + \)\(51\!\cdots\!76\)\( p^{124} T^{6} - \)\(23\!\cdots\!80\)\( p^{184} T^{7} + p^{244} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(32\!\cdots\!28\)\( T + \)\(28\!\cdots\!68\)\( T^{2} - \)\(24\!\cdots\!96\)\( p T^{3} + \)\(39\!\cdots\!70\)\( p^{2} T^{4} - \)\(24\!\cdots\!96\)\( p^{62} T^{5} + \)\(28\!\cdots\!68\)\( p^{122} T^{6} - \)\(32\!\cdots\!28\)\( p^{183} T^{7} + p^{244} T^{8} \)
37$C_2 \wr S_4$ \( 1 + \)\(71\!\cdots\!00\)\( T + \)\(26\!\cdots\!80\)\( p T^{2} + \)\(49\!\cdots\!00\)\( p^{2} T^{3} + \)\(13\!\cdots\!46\)\( p^{3} T^{4} + \)\(49\!\cdots\!00\)\( p^{63} T^{5} + \)\(26\!\cdots\!80\)\( p^{123} T^{6} + \)\(71\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(16\!\cdots\!68\)\( T + \)\(19\!\cdots\!28\)\( p T^{2} - \)\(50\!\cdots\!36\)\( p^{2} T^{3} + \)\(38\!\cdots\!70\)\( p^{3} T^{4} - \)\(50\!\cdots\!36\)\( p^{63} T^{5} + \)\(19\!\cdots\!28\)\( p^{123} T^{6} - \)\(16\!\cdots\!68\)\( p^{183} T^{7} + p^{244} T^{8} \)
43$C_2 \wr S_4$ \( 1 - \)\(75\!\cdots\!00\)\( T + \)\(31\!\cdots\!00\)\( p T^{2} - \)\(43\!\cdots\!00\)\( p^{2} T^{3} + \)\(10\!\cdots\!14\)\( p^{3} T^{4} - \)\(43\!\cdots\!00\)\( p^{63} T^{5} + \)\(31\!\cdots\!00\)\( p^{123} T^{6} - \)\(75\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(21\!\cdots\!00\)\( T + \)\(77\!\cdots\!40\)\( p T^{2} + \)\(20\!\cdots\!00\)\( p^{2} T^{3} + \)\(52\!\cdots\!66\)\( p^{3} T^{4} + \)\(20\!\cdots\!00\)\( p^{63} T^{5} + \)\(77\!\cdots\!40\)\( p^{123} T^{6} + \)\(21\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(30\!\cdots\!00\)\( p^{2} T + \)\(19\!\cdots\!20\)\( p^{2} T^{2} - \)\(13\!\cdots\!00\)\( p^{3} T^{3} + \)\(11\!\cdots\!78\)\( p^{4} T^{4} - \)\(13\!\cdots\!00\)\( p^{64} T^{5} + \)\(19\!\cdots\!20\)\( p^{124} T^{6} - \)\(30\!\cdots\!00\)\( p^{185} T^{7} + p^{244} T^{8} \)
59$C_2 \wr S_4$ \( 1 + \)\(36\!\cdots\!40\)\( p T + \)\(10\!\cdots\!56\)\( p^{2} T^{2} + \)\(24\!\cdots\!80\)\( p^{3} T^{3} + \)\(51\!\cdots\!26\)\( p^{4} T^{4} + \)\(24\!\cdots\!80\)\( p^{64} T^{5} + \)\(10\!\cdots\!56\)\( p^{124} T^{6} + \)\(36\!\cdots\!40\)\( p^{184} T^{7} + p^{244} T^{8} \)
61$C_2 \wr S_4$ \( 1 - \)\(42\!\cdots\!48\)\( T + \)\(51\!\cdots\!28\)\( p T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!96\)\( p^{61} T^{5} + \)\(51\!\cdots\!28\)\( p^{123} T^{6} - \)\(42\!\cdots\!48\)\( p^{183} T^{7} + p^{244} T^{8} \)
67$C_2 \wr S_4$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(14\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!78\)\( T^{4} + \)\(10\!\cdots\!00\)\( p^{61} T^{5} + \)\(14\!\cdots\!20\)\( p^{122} T^{6} + \)\(15\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
71$C_2 \wr S_4$ \( 1 - \)\(26\!\cdots\!88\)\( T + \)\(15\!\cdots\!88\)\( T^{2} + \)\(22\!\cdots\!64\)\( T^{3} + \)\(64\!\cdots\!70\)\( T^{4} + \)\(22\!\cdots\!64\)\( p^{61} T^{5} + \)\(15\!\cdots\!88\)\( p^{122} T^{6} - \)\(26\!\cdots\!88\)\( p^{183} T^{7} + p^{244} T^{8} \)
73$C_2 \wr S_4$ \( 1 - \)\(43\!\cdots\!00\)\( T + \)\(13\!\cdots\!40\)\( T^{2} - \)\(45\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!58\)\( T^{4} - \)\(45\!\cdots\!00\)\( p^{61} T^{5} + \)\(13\!\cdots\!40\)\( p^{122} T^{6} - \)\(43\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
79$C_2 \wr S_4$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(11\!\cdots\!16\)\( T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!46\)\( T^{4} - \)\(36\!\cdots\!40\)\( p^{61} T^{5} + \)\(11\!\cdots\!16\)\( p^{122} T^{6} + \)\(21\!\cdots\!80\)\( p^{183} T^{7} + p^{244} T^{8} \)
83$C_2 \wr S_4$ \( 1 + \)\(19\!\cdots\!00\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(86\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!78\)\( T^{4} + \)\(86\!\cdots\!00\)\( p^{61} T^{5} + \)\(16\!\cdots\!20\)\( p^{122} T^{6} + \)\(19\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(60\!\cdots\!40\)\( T + \)\(42\!\cdots\!56\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!26\)\( T^{4} + \)\(15\!\cdots\!80\)\( p^{61} T^{5} + \)\(42\!\cdots\!56\)\( p^{122} T^{6} + \)\(60\!\cdots\!40\)\( p^{183} T^{7} + p^{244} T^{8} \)
97$C_2 \wr S_4$ \( 1 + \)\(80\!\cdots\!00\)\( T + \)\(80\!\cdots\!80\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!18\)\( T^{4} + \)\(38\!\cdots\!00\)\( p^{61} T^{5} + \)\(80\!\cdots\!80\)\( p^{122} T^{6} + \)\(80\!\cdots\!00\)\( p^{183} T^{7} + p^{244} T^{8} \)
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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

−13.26463982769579384643463080700, −12.21392220019893156749168549316, −12.17130866628167511468589375964, −11.44898793875203414420182295781, −11.22461445873856023968408214686, −10.75206702783945454322899220034, −10.17967125104493943940457901527, −9.613615850869011209910268229198, −9.360498338412850319100496424700, −8.435081671963490783505478108991, −8.035937676701038507713518680136, −7.86762902956605463769700265963, −6.70471629149157498504648306267, −6.52148749637659767175979247396, −5.80335275982875559328762633248, −5.61929033798567742138441997144, −5.58693134966324093625404413917, −4.63481380677916661648876098095, −4.21345880439988315458369248470, −3.97058657026526966063947217473, −3.57519995933766153820583028905, −2.72237056783497296800658086066, −2.41181351419019483043370514240, −1.56757984594175452865044515812, −1.23948223022818828095323826042, 0, 0, 0, 0, 1.23948223022818828095323826042, 1.56757984594175452865044515812, 2.41181351419019483043370514240, 2.72237056783497296800658086066, 3.57519995933766153820583028905, 3.97058657026526966063947217473, 4.21345880439988315458369248470, 4.63481380677916661648876098095, 5.58693134966324093625404413917, 5.61929033798567742138441997144, 5.80335275982875559328762633248, 6.52148749637659767175979247396, 6.70471629149157498504648306267, 7.86762902956605463769700265963, 8.035937676701038507713518680136, 8.435081671963490783505478108991, 9.360498338412850319100496424700, 9.613615850869011209910268229198, 10.17967125104493943940457901527, 10.75206702783945454322899220034, 11.22461445873856023968408214686, 11.44898793875203414420182295781, 12.17130866628167511468589375964, 12.21392220019893156749168549316, 13.26463982769579384643463080700

Graph of the $Z$-function along the critical line