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Properties

Label 8-50e4-1.1-c23e4-0-0
Degree $8$
Conductor $6250000$
Sign $1$
Analytic cond. $7.89072\times 10^{8}$
Root an. cond. $12.9461$
Motivic weight $23$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.38e6·4-s + 2.00e11·9-s − 1.38e12·11-s + 5.27e13·16-s + 1.94e15·19-s + 8.07e16·29-s − 1.09e17·31-s − 1.68e18·36-s − 8.54e18·41-s + 1.16e19·44-s + 6.48e19·49-s − 8.03e19·59-s + 1.60e21·61-s − 2.95e20·64-s + 1.43e21·71-s − 1.62e22·76-s + 1.41e22·79-s + 1.31e22·81-s + 3.96e20·89-s − 2.77e23·99-s + 9.33e22·101-s + 1.20e24·109-s − 6.77e23·116-s − 2.23e24·121-s + 9.19e23·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 2.12·9-s − 1.46·11-s + 3/4·16-s + 3.82·19-s + 1.22·29-s − 0.774·31-s − 2.12·36-s − 2.42·41-s + 1.46·44-s + 2.37·49-s − 0.347·59-s + 4.72·61-s − 1/2·64-s + 0.737·71-s − 3.82·76-s + 2.12·79-s + 1.48·81-s + 0.0151·89-s − 3.11·99-s + 0.832·101-s + 4.48·109-s − 1.22·116-s − 2.49·121-s + 0.774·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+23/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6250000\)    =    \(2^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(7.89072\times 10^{8}\)
Root analytic conductor: \(12.9461\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6250000,\ (\ :23/2, 23/2, 23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(0.7733313634\)
\(L(\frac12)\) \(\approx\) \(0.7733313634\)
\(L(\frac{25}{2})\) 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)$
bad2$C_2$ \( ( 1 + p^{22} T^{2} )^{2} \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - 22271434820 p^{2} T^{2} + 458001548506188742 p^{10} T^{4} - 22271434820 p^{48} T^{6} + p^{92} T^{8} \)
7$D_4\times C_2$ \( 1 - 1323849609095360660 p^{2} T^{2} + \)\(20\!\cdots\!02\)\( p^{6} T^{4} - 1323849609095360660 p^{48} T^{6} + p^{92} T^{8} \)
11$D_{4}$ \( ( 1 + 62935415616 p T + \)\(15\!\cdots\!86\)\( p^{2} T^{2} + 62935415616 p^{24} T^{3} + p^{46} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - \)\(25\!\cdots\!60\)\( T^{2} + \)\(20\!\cdots\!22\)\( p^{2} T^{4} - \)\(25\!\cdots\!60\)\( p^{46} T^{6} + p^{92} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(63\!\cdots\!40\)\( T^{2} + \)\(61\!\cdots\!42\)\( p^{2} T^{4} - \)\(63\!\cdots\!40\)\( p^{46} T^{6} + p^{92} T^{8} \)
19$D_{4}$ \( ( 1 - 970438534616600 T + \)\(38\!\cdots\!22\)\( p T^{2} - 970438534616600 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(62\!\cdots\!20\)\( T^{2} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(62\!\cdots\!20\)\( p^{46} T^{6} + p^{92} T^{8} \)
29$D_{4}$ \( ( 1 - 40394756391690180 T + \)\(26\!\cdots\!78\)\( T^{2} - 40394756391690180 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 54819002015519096 T + \)\(33\!\cdots\!86\)\( T^{2} + 54819002015519096 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(18\!\cdots\!00\)\( T^{2} + \)\(23\!\cdots\!18\)\( T^{4} - \)\(18\!\cdots\!00\)\( p^{46} T^{6} + p^{92} T^{8} \)
41$D_{4}$ \( ( 1 + 4273739814696341076 T + \)\(22\!\cdots\!86\)\( T^{2} + 4273739814696341076 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(80\!\cdots\!80\)\( T^{2} + \)\(38\!\cdots\!98\)\( T^{4} - \)\(80\!\cdots\!80\)\( p^{46} T^{6} + p^{92} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(34\!\cdots\!60\)\( T^{2} + \)\(32\!\cdots\!58\)\( T^{4} - \)\(34\!\cdots\!60\)\( p^{46} T^{6} + p^{92} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!40\)\( T^{2} + \)\(64\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!40\)\( p^{46} T^{6} + p^{92} T^{8} \)
59$D_{4}$ \( ( 1 + 40199521927491576840 T + \)\(66\!\cdots\!58\)\( T^{2} + 40199521927491576840 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - \)\(80\!\cdots\!04\)\( T + \)\(37\!\cdots\!66\)\( T^{2} - \)\(80\!\cdots\!04\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!00\)\( T^{2} + \)\(17\!\cdots\!38\)\( T^{4} - \)\(14\!\cdots\!00\)\( p^{46} T^{6} + p^{92} T^{8} \)
71$D_{4}$ \( ( 1 - \)\(71\!\cdots\!64\)\( T - \)\(15\!\cdots\!54\)\( T^{2} - \)\(71\!\cdots\!64\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(82\!\cdots\!80\)\( T^{2} + \)\(43\!\cdots\!78\)\( T^{4} - \)\(82\!\cdots\!80\)\( p^{46} T^{6} + p^{92} T^{8} \)
79$D_{4}$ \( ( 1 - \)\(70\!\cdots\!20\)\( T + \)\(10\!\cdots\!78\)\( T^{2} - \)\(70\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(17\!\cdots\!80\)\( T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - \)\(17\!\cdots\!80\)\( p^{46} T^{6} + p^{92} T^{8} \)
89$D_{4}$ \( ( 1 - \)\(19\!\cdots\!80\)\( T + \)\(11\!\cdots\!38\)\( T^{2} - \)\(19\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(72\!\cdots\!20\)\( T^{2} + \)\(22\!\cdots\!58\)\( T^{4} - \)\(72\!\cdots\!20\)\( p^{46} T^{6} + p^{92} T^{8} \)
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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

−7.26267989440015201953483595839, −7.13604840584669986906481464201, −7.12509578474095856742407051083, −6.38558577042684424347480818659, −6.28342854487835865733797376851, −5.61245033721831569276492278109, −5.50023356415719037548480259330, −5.16056459698266726918008805581, −4.91063633745120909202895613576, −4.85980729291180296776594604691, −4.68273509971737381633048310653, −3.80144027922060055569715841749, −3.72074234548431133137519782365, −3.64688561076860561966045627701, −3.55381115303999432463981783678, −2.75033803246208929373718409054, −2.68844007942149560517496901297, −2.24880426914318431734225795638, −2.11637932776584669049868861872, −1.39185870470309711112928146636, −1.29979859032035994580253769868, −1.05606265038972972883681836332, −0.842146679937357100442094222784, −0.61085242407239084228608677716, −0.07813250672482166511139869778, 0.07813250672482166511139869778, 0.61085242407239084228608677716, 0.842146679937357100442094222784, 1.05606265038972972883681836332, 1.29979859032035994580253769868, 1.39185870470309711112928146636, 2.11637932776584669049868861872, 2.24880426914318431734225795638, 2.68844007942149560517496901297, 2.75033803246208929373718409054, 3.55381115303999432463981783678, 3.64688561076860561966045627701, 3.72074234548431133137519782365, 3.80144027922060055569715841749, 4.68273509971737381633048310653, 4.85980729291180296776594604691, 4.91063633745120909202895613576, 5.16056459698266726918008805581, 5.50023356415719037548480259330, 5.61245033721831569276492278109, 6.28342854487835865733797376851, 6.38558577042684424347480818659, 7.12509578474095856742407051083, 7.13604840584669986906481464201, 7.26267989440015201953483595839

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