Properties

Label 8-75e4-1.1-c2e4-0-1
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $17.4415$
Root an. cond. $1.42954$
Motivic weight $2$
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  − 6·4-s + 2·9-s − 5·16-s + 8·19-s − 72·31-s − 12·36-s + 124·49-s + 328·61-s + 180·64-s − 48·76-s − 552·79-s − 77·81-s − 152·109-s + 444·121-s + 432·124-s + ⋯
L(s)  = 1  − 3/2·4-s + 2/9·9-s − 0.312·16-s + 8/19·19-s − 2.32·31-s − 1/3·36-s + 2.53·49-s + 5.37·61-s + 2.81·64-s − 0.631·76-s − 6.98·79-s − 0.950·81-s − 1.39·109-s + 3.66·121-s + 3.48·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(17.4415\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8183758871\)
\(L(\frac12)\) \(\approx\) \(0.8183758871\)
\(L(2)\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
5 \( 1 \)
good2$C_2^2$ \( ( 1 + 3 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 222 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 82 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 558 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 878 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 702 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 558 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 3442 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 1998 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6942 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 8402 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 5182 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 138 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 4958 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( ( 1 + 8738 T^{2} + p^{4} T^{4} )^{2} \)
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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

−10.42312508423581756441210393947, −9.957489967599238255946509188327, −9.918549321675346939345970827289, −9.835576344788790113193440943264, −9.347234988067112550107610244410, −8.840780556043446652023125369066, −8.695916637809971321562132067150, −8.594793722974099512644820849458, −8.465249663615244286862819829368, −7.66099038606770347015452052937, −7.27020219943647043171970581362, −7.24552569594067322114938871160, −6.88725113423520709060004447181, −6.46956883220554981972872824927, −5.69209964193781473695484719826, −5.55406273432819310902086409115, −5.39194695345011453929921597559, −4.87465527927019943797374885879, −4.24628357171964507243442624048, −4.00876422713434947061335680192, −3.96647598190035573314713142173, −3.07297585787477368925610568184, −2.48672149529654840341633430622, −1.76146658001021838191090524086, −0.56232865399743246886736453768, 0.56232865399743246886736453768, 1.76146658001021838191090524086, 2.48672149529654840341633430622, 3.07297585787477368925610568184, 3.96647598190035573314713142173, 4.00876422713434947061335680192, 4.24628357171964507243442624048, 4.87465527927019943797374885879, 5.39194695345011453929921597559, 5.55406273432819310902086409115, 5.69209964193781473695484719826, 6.46956883220554981972872824927, 6.88725113423520709060004447181, 7.24552569594067322114938871160, 7.27020219943647043171970581362, 7.66099038606770347015452052937, 8.465249663615244286862819829368, 8.594793722974099512644820849458, 8.695916637809971321562132067150, 8.840780556043446652023125369066, 9.347234988067112550107610244410, 9.835576344788790113193440943264, 9.918549321675346939345970827289, 9.957489967599238255946509188327, 10.42312508423581756441210393947

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