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Properties

Label 8-50e4-1.1-c25e4-0-0
Degree $8$
Conductor $6250000$
Sign $1$
Analytic cond. $1.53690\times 10^{9}$
Root an. cond. $14.0711$
Motivic weight $25$
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  − 3.35e7·4-s + 1.34e12·9-s − 1.67e13·11-s + 8.44e14·16-s − 6.97e15·19-s + 1.07e18·29-s − 7.24e18·31-s − 4.50e19·36-s − 4.25e20·41-s + 5.62e20·44-s + 3.82e21·49-s + 2.62e22·59-s + 2.28e20·61-s − 1.88e22·64-s + 1.67e23·71-s + 2.34e23·76-s − 2.11e23·79-s + 1.95e23·81-s + 3.75e24·89-s − 2.24e25·99-s − 4.00e25·101-s − 1.33e26·109-s − 3.60e25·116-s − 2.11e25·121-s + 2.43e26·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 1.58·9-s − 1.60·11-s + 3/4·16-s − 0.723·19-s + 0.564·29-s − 1.65·31-s − 1.58·36-s − 2.94·41-s + 1.60·44-s + 2.84·49-s + 1.92·59-s + 0.0110·61-s − 1/2·64-s + 1.21·71-s + 0.723·76-s − 0.403·79-s + 0.272·81-s + 1.61·89-s − 2.54·99-s − 3.54·101-s − 4.54·109-s − 0.564·116-s − 0.194·121-s + 1.65·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+25/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6250000\)    =    \(2^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.53690\times 10^{9}\)
Root analytic conductor: \(14.0711\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6250000,\ (\ :25/2, 25/2, 25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(0.1012403369\)
\(L(\frac12)\) \(\approx\) \(0.1012403369\)
\(L(\frac{27}{2})\) 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$C_2$ \( ( 1 + p^{24} T^{2} )^{2} \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - 1341166746500 T^{2} + \)\(21\!\cdots\!62\)\( p^{6} T^{4} - 1341166746500 p^{50} T^{6} + p^{100} T^{8} \)
7$D_4\times C_2$ \( 1 - 77979681564870849620 p^{2} T^{2} + \)\(11\!\cdots\!98\)\( p^{8} T^{4} - 77979681564870849620 p^{52} T^{6} + p^{100} T^{8} \)
11$D_{4}$ \( ( 1 + 8379169876416 T + \)\(95\!\cdots\!46\)\( p^{2} T^{2} + 8379169876416 p^{25} T^{3} + p^{50} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - \)\(41\!\cdots\!00\)\( p^{2} T^{2} + \)\(20\!\cdots\!18\)\( p^{4} T^{4} - \)\(41\!\cdots\!00\)\( p^{52} T^{6} + p^{100} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(17\!\cdots\!40\)\( T^{2} + \)\(47\!\cdots\!82\)\( p^{2} T^{4} - \)\(17\!\cdots\!40\)\( p^{50} T^{6} + p^{100} T^{8} \)
19$D_{4}$ \( ( 1 + 183655863992680 p T + \)\(15\!\cdots\!18\)\( p^{2} T^{2} + 183655863992680 p^{26} T^{3} + p^{50} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(62\!\cdots\!40\)\( p^{2} T^{2} + \)\(17\!\cdots\!78\)\( p^{4} T^{4} - \)\(62\!\cdots\!40\)\( p^{52} T^{6} + p^{100} T^{8} \)
29$D_{4}$ \( ( 1 - 537443787959856180 T + \)\(60\!\cdots\!98\)\( T^{2} - 537443787959856180 p^{25} T^{3} + p^{50} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 3624027784441312136 T + \)\(39\!\cdots\!26\)\( T^{2} + 3624027784441312136 p^{25} T^{3} + p^{50} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + \)\(36\!\cdots\!20\)\( T^{2} + \)\(40\!\cdots\!98\)\( T^{4} + \)\(36\!\cdots\!20\)\( p^{50} T^{6} + p^{100} T^{8} \)
41$D_{4}$ \( ( 1 + \)\(21\!\cdots\!56\)\( T + \)\(52\!\cdots\!86\)\( T^{2} + \)\(21\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(24\!\cdots\!60\)\( T^{2} + \)\(24\!\cdots\!98\)\( T^{4} - \)\(24\!\cdots\!60\)\( p^{50} T^{6} + p^{100} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(11\!\cdots\!60\)\( p^{50} T^{6} + p^{100} T^{8} \)
53$D_4\times C_2$ \( 1 + \)\(51\!\cdots\!80\)\( T^{2} + \)\(89\!\cdots\!98\)\( T^{4} + \)\(51\!\cdots\!80\)\( p^{50} T^{6} + p^{100} T^{8} \)
59$D_{4}$ \( ( 1 - \)\(13\!\cdots\!60\)\( T + \)\(39\!\cdots\!98\)\( T^{2} - \)\(13\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - \)\(11\!\cdots\!24\)\( T - \)\(13\!\cdots\!54\)\( T^{2} - \)\(11\!\cdots\!24\)\( p^{25} T^{3} + p^{50} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(13\!\cdots\!60\)\( T^{2} + \)\(85\!\cdots\!98\)\( T^{4} - \)\(13\!\cdots\!60\)\( p^{50} T^{6} + p^{100} T^{8} \)
71$D_{4}$ \( ( 1 - \)\(83\!\cdots\!24\)\( T + \)\(34\!\cdots\!46\)\( T^{2} - \)\(83\!\cdots\!24\)\( p^{25} T^{3} + p^{50} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(49\!\cdots\!80\)\( T^{2} + \)\(18\!\cdots\!98\)\( T^{4} - \)\(49\!\cdots\!80\)\( p^{50} T^{6} + p^{100} T^{8} \)
79$D_{4}$ \( ( 1 + \)\(10\!\cdots\!40\)\( T + \)\(51\!\cdots\!98\)\( T^{2} + \)\(10\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(82\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - \)\(82\!\cdots\!20\)\( p^{50} T^{6} + p^{100} T^{8} \)
89$D_{4}$ \( ( 1 - \)\(18\!\cdots\!20\)\( T + \)\(11\!\cdots\!98\)\( T^{2} - \)\(18\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!80\)\( T^{2} + \)\(75\!\cdots\!98\)\( T^{4} - \)\(11\!\cdots\!80\)\( p^{50} T^{6} + p^{100} T^{8} \)
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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

−6.95679904114462605371394553445, −6.94941453932342488415563784546, −6.86967683782969449094406017182, −6.31513769984505748271289946227, −5.83591937237979454060190926086, −5.50346991322674737867307985902, −5.48265172401808364414376458014, −5.04569103556410237178510970431, −4.92437090223178012415382602729, −4.65330664341459574898701971695, −4.19711503533296956641994840307, −3.83475545778180951225738188826, −3.78407562616293340657949270870, −3.75681242092132144833573822475, −2.97046463722268900904399722620, −2.94042150700405804658016344491, −2.27628633879304211092293677414, −2.24813025760978642019434383754, −2.14999676280918708431829459202, −1.41903913306791764844112626345, −1.25519556630877727280724094134, −1.15662445306369396906185232805, −0.816632847912762244215606200004, −0.11904702918877395204527773977, −0.11585473837511834244886636408, 0.11585473837511834244886636408, 0.11904702918877395204527773977, 0.816632847912762244215606200004, 1.15662445306369396906185232805, 1.25519556630877727280724094134, 1.41903913306791764844112626345, 2.14999676280918708431829459202, 2.24813025760978642019434383754, 2.27628633879304211092293677414, 2.94042150700405804658016344491, 2.97046463722268900904399722620, 3.75681242092132144833573822475, 3.78407562616293340657949270870, 3.83475545778180951225738188826, 4.19711503533296956641994840307, 4.65330664341459574898701971695, 4.92437090223178012415382602729, 5.04569103556410237178510970431, 5.48265172401808364414376458014, 5.50346991322674737867307985902, 5.83591937237979454060190926086, 6.31513769984505748271289946227, 6.86967683782969449094406017182, 6.94941453932342488415563784546, 6.95679904114462605371394553445

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