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Properties

Label 8-75e4-1.1-c12e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $2.20809\times 10^{7}$
Root an. cond. $8.27946$
Motivic weight $12$
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  + 464·4-s − 7.59e5·9-s − 3.33e7·16-s − 2.13e8·19-s + 2.66e8·31-s − 3.52e8·36-s + 5.21e10·49-s − 1.62e11·61-s − 2.33e10·64-s − 9.90e10·76-s + 1.00e12·79-s + 2.94e11·81-s − 1.08e13·109-s + 9.85e12·121-s + 1.23e11·124-s + 127-s + 131-s + 137-s + 139-s + 2.53e13·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.98e13·169-s + 1.62e14·171-s + ⋯
L(s)  = 1  + 0.113·4-s − 1.42·9-s − 1.99·16-s − 4.53·19-s + 0.299·31-s − 0.161·36-s + 3.76·49-s − 3.15·61-s − 0.339·64-s − 0.513·76-s + 4.15·79-s + 1.04·81-s − 6.47·109-s + 3.14·121-s + 0.0339·124-s + 2.84·144-s + 3.85·169-s + 6.48·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.20809\times 10^{7}\)
Root analytic conductor: \(8.27946\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :6, 6, 6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.9985535571\)
\(L(\frac12)\) \(\approx\) \(0.9985535571\)
\(L(7)\) 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$C_2^2$ \( 1 + 1042 p^{6} T^{2} + p^{24} T^{4} \)
5 \( 1 \)
good2$C_2^2$ \( ( 1 - 29 p^{3} T^{2} + p^{24} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 531887998 p^{2} T^{2} + p^{24} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 40734109202 p^{2} T^{2} + p^{24} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 44947385842462 T^{2} + p^{24} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 944884986604418 T^{2} + p^{24} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 53343578 T + p^{12} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 61021177942562 p^{2} T^{2} + p^{24} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 693172036445878082 T^{2} + p^{24} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 66526202 T + p^{12} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 5988811119613198 p^{2} T^{2} + p^{24} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 22304779456187067838 T^{2} + p^{24} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 671150004738909698 T^{2} + p^{24} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + \)\(23\!\cdots\!78\)\( T^{2} + p^{24} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - \)\(71\!\cdots\!22\)\( T^{2} + p^{24} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - \)\(14\!\cdots\!62\)\( T^{2} + p^{24} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 40679935918 T + p^{12} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - \)\(16\!\cdots\!22\)\( T^{2} + p^{24} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - \)\(30\!\cdots\!82\)\( T^{2} + p^{24} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - \)\(42\!\cdots\!42\)\( T^{2} + p^{24} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 252324997702 T + p^{12} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + \)\(45\!\cdots\!38\)\( T^{2} + p^{24} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - \)\(48\!\cdots\!42\)\( T^{2} + p^{24} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - \)\(96\!\cdots\!82\)\( T^{2} + p^{24} T^{4} )^{2} \)
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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

−8.240992812943093139133447533070, −8.099729580180423926136497282373, −7.82475036763747860740529747104, −7.26004965146260167397332494995, −6.91757466252133178290726185456, −6.54829216258847855452420280808, −6.42861059280173388839767312868, −6.35576900169028651753722948259, −5.72695316724097913707361684797, −5.57144783232880996299868048949, −5.21361284565541345576224283311, −4.61353431200890362551071518281, −4.33623556242560192959442615579, −4.18651447283822811675516337966, −4.11646337641420350815442192106, −3.28976115512873308701559259298, −3.13385180473722374498553261487, −2.45855668673511062852868229416, −2.35193822738425791519248897724, −2.18127333970212149685494714757, −1.93003837311573836504831850566, −1.34871400727366069129393900768, −0.75338729612149116043933954556, −0.33209265521488702333430184545, −0.23339234915815577210636806543, 0.23339234915815577210636806543, 0.33209265521488702333430184545, 0.75338729612149116043933954556, 1.34871400727366069129393900768, 1.93003837311573836504831850566, 2.18127333970212149685494714757, 2.35193822738425791519248897724, 2.45855668673511062852868229416, 3.13385180473722374498553261487, 3.28976115512873308701559259298, 4.11646337641420350815442192106, 4.18651447283822811675516337966, 4.33623556242560192959442615579, 4.61353431200890362551071518281, 5.21361284565541345576224283311, 5.57144783232880996299868048949, 5.72695316724097913707361684797, 6.35576900169028651753722948259, 6.42861059280173388839767312868, 6.54829216258847855452420280808, 6.91757466252133178290726185456, 7.26004965146260167397332494995, 7.82475036763747860740529747104, 8.099729580180423926136497282373, 8.240992812943093139133447533070

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