Properties

Label 8-75e4-1.1-c13e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $4.18336\times 10^{7}$
Root an. cond. $8.96789$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.01e4·4-s − 1.06e6·9-s + 1.24e7·11-s + 6.34e7·16-s − 7.69e7·19-s − 2.25e9·29-s + 1.22e10·31-s − 1.07e10·36-s − 5.92e10·41-s + 1.26e11·44-s + 2.27e11·49-s − 1.44e12·59-s − 4.59e11·61-s + 9.24e11·64-s + 1.47e12·71-s − 7.80e11·76-s − 7.05e12·79-s + 8.47e11·81-s − 1.18e13·89-s − 1.32e13·99-s − 2.75e13·101-s + 1.02e13·109-s − 2.28e13·116-s − 2.28e13·121-s + 1.23e14·124-s + ⋯
L(s)  = 1  + 1.23·4-s − 2/3·9-s + 2.12·11-s + 0.945·16-s − 0.375·19-s − 0.703·29-s + 2.46·31-s − 0.825·36-s − 1.94·41-s + 2.63·44-s + 2.34·49-s − 4.46·59-s − 1.14·61-s + 1.68·64-s + 1.36·71-s − 0.464·76-s − 3.26·79-s + 1/3·81-s − 2.52·89-s − 1.41·99-s − 2.58·101-s + 0.585·109-s − 0.870·116-s − 0.660·121-s + 3.05·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(4.18336\times 10^{7}\)
Root analytic conductor: \(8.96789\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(0.4798152309\)
\(L(\frac12)\) \(\approx\) \(0.4798152309\)
\(L(\frac{15}{2})\) 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)$
bad3$C_2$ \( ( 1 + p^{12} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 - 10143 T^{2} + 2464313 p^{4} T^{4} - 10143 p^{26} T^{6} + p^{52} T^{8} \)
7$D_4\times C_2$ \( 1 - 4640254396 p^{2} T^{2} + 11302885510790062758 p^{4} T^{4} - 4640254396 p^{28} T^{6} + p^{52} T^{8} \)
11$D_{4}$ \( ( 1 - 567808 p T + 577871634854 p^{2} T^{2} - 567808 p^{14} T^{3} + p^{26} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 1209821824684172 T^{2} + \)\(54\!\cdots\!98\)\( T^{4} - 1209821824684172 p^{26} T^{6} + p^{52} T^{8} \)
17$D_4\times C_2$ \( 1 - 33243340715717340 T^{2} + \)\(46\!\cdots\!38\)\( T^{4} - 33243340715717340 p^{26} T^{6} + p^{52} T^{8} \)
19$D_{4}$ \( ( 1 + 38475344 T - 37450960045515402 T^{2} + 38475344 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 1725325536935916956 T^{2} + \)\(12\!\cdots\!18\)\( T^{4} - 1725325536935916956 p^{26} T^{6} + p^{52} T^{8} \)
29$D_{4}$ \( ( 1 + 1126474708 T - 2318209037061176962 T^{2} + 1126474708 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 6100104312 T + 36788978328798625982 T^{2} - 6100104312 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + \)\(16\!\cdots\!56\)\( T^{2} + \)\(73\!\cdots\!18\)\( T^{4} + \)\(16\!\cdots\!56\)\( p^{26} T^{6} + p^{52} T^{8} \)
41$D_{4}$ \( ( 1 + 29607909788 T + \)\(20\!\cdots\!22\)\( T^{2} + 29607909788 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(20\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!20\)\( p^{26} T^{6} + p^{52} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!20\)\( T^{2} + \)\(66\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!20\)\( p^{26} T^{6} + p^{52} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(18\!\cdots\!36\)\( T^{2} + \)\(13\!\cdots\!98\)\( T^{4} - \)\(18\!\cdots\!36\)\( p^{26} T^{6} + p^{52} T^{8} \)
59$D_{4}$ \( ( 1 + 722768180096 T + \)\(33\!\cdots\!58\)\( T^{2} + 722768180096 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 229984402180 T + \)\(25\!\cdots\!38\)\( T^{2} + 229984402180 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!20\)\( T^{2} + \)\(17\!\cdots\!38\)\( T^{4} - \)\(21\!\cdots\!20\)\( p^{26} T^{6} + p^{52} T^{8} \)
71$D_{4}$ \( ( 1 - 736545668224 T + \)\(18\!\cdots\!66\)\( T^{2} - 736545668224 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(47\!\cdots\!76\)\( T^{2} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(47\!\cdots\!76\)\( p^{26} T^{6} + p^{52} T^{8} \)
79$D_{4}$ \( ( 1 + 3526348590120 T + \)\(74\!\cdots\!78\)\( T^{2} + 3526348590120 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!92\)\( T^{2} + \)\(71\!\cdots\!78\)\( T^{4} - \)\(11\!\cdots\!92\)\( p^{26} T^{6} + p^{52} T^{8} \)
89$D_{4}$ \( ( 1 + 5913121459764 T + \)\(37\!\cdots\!18\)\( T^{2} + 5913121459764 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(70\!\cdots\!40\)\( T^{2} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(70\!\cdots\!40\)\( p^{26} T^{6} + p^{52} T^{8} \)
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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

−7.899896740003746413974808595213, −7.72599925172839307715741174931, −7.68425988204786963567656359440, −6.83704471929601183386814029911, −6.79443802620578768482689767532, −6.61364213727169852285794522980, −6.54970489048092819212639070805, −5.82466493585038135965767348805, −5.72284449142116057655195882781, −5.66839168407970845616288081687, −4.88350077426970493848136519226, −4.67081400667340329317541643082, −4.26705515107946430087690250375, −3.87999441895081321917214534400, −3.84544352040519872753275946485, −3.12004888419063965202318476419, −2.94242180875498695624073754029, −2.75354946860234905761544140373, −2.40100444015396141406161202151, −1.67723418061361895352273253998, −1.64640333909182783755780972088, −1.45180379714837819748254079063, −0.993421438538933513090786696459, −0.65004998814356672619263685104, −0.06307403620646684077692587361, 0.06307403620646684077692587361, 0.65004998814356672619263685104, 0.993421438538933513090786696459, 1.45180379714837819748254079063, 1.64640333909182783755780972088, 1.67723418061361895352273253998, 2.40100444015396141406161202151, 2.75354946860234905761544140373, 2.94242180875498695624073754029, 3.12004888419063965202318476419, 3.84544352040519872753275946485, 3.87999441895081321917214534400, 4.26705515107946430087690250375, 4.67081400667340329317541643082, 4.88350077426970493848136519226, 5.66839168407970845616288081687, 5.72284449142116057655195882781, 5.82466493585038135965767348805, 6.54970489048092819212639070805, 6.61364213727169852285794522980, 6.79443802620578768482689767532, 6.83704471929601183386814029911, 7.68425988204786963567656359440, 7.72599925172839307715741174931, 7.899896740003746413974808595213

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