Properties

Label 8-370e4-1.1-c1e4-0-0
Degree $8$
Conductor $18741610000$
Sign $1$
Analytic cond. $76.1930$
Root an. cond. $1.71885$
Motivic weight $1$
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  − 2·4-s − 8·5-s + 3·16-s − 24·19-s + 16·20-s + 38·25-s − 16·31-s + 8·41-s + 8·49-s + 24·59-s + 48·61-s − 4·64-s + 48·76-s − 16·79-s − 24·80-s − 18·81-s − 24·89-s + 192·95-s − 76·100-s − 16·101-s − 16·109-s + 4·121-s + 32·124-s − 136·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 4-s − 3.57·5-s + 3/4·16-s − 5.50·19-s + 3.57·20-s + 38/5·25-s − 2.87·31-s + 1.24·41-s + 8/7·49-s + 3.12·59-s + 6.14·61-s − 1/2·64-s + 5.50·76-s − 1.80·79-s − 2.68·80-s − 2·81-s − 2.54·89-s + 19.6·95-s − 7.59·100-s − 1.59·101-s − 1.53·109-s + 4/11·121-s + 2.87·124-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(76.1930\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1129772755\)
\(L(\frac12)\) \(\approx\) \(0.1129772755\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 8 T^{2} + 18 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 12 T^{2} - 442 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$D_{4}$ \( ( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 168 T^{2} + 11378 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 12 T^{2} + 4118 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 12 T + 148 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
67$D_4\times C_2$ \( 1 - 128 T^{2} + 11538 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 32 T^{2} + 10578 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \)
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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

−8.370542814523166876391236218295, −8.130256828769222925289323381373, −8.126851331913683768832510524418, −7.28121656645077669305103295886, −7.26878216852945703828624592089, −7.23217428890444396756726139354, −6.94772658743868467461326405431, −6.55394008362429058527231465249, −6.47276140413716617276303367793, −5.95292952459366795557344906314, −5.43810859881776782190466656253, −5.43272995429474713041559300793, −5.12846248971825754738029212440, −4.57075871711358165675726411790, −4.30837356157822123939895533126, −4.00228210919911626960026921576, −3.91829740367064132770064468083, −3.90734779431379043345770183706, −3.90565565502326318883555417914, −3.04726820625125862859212933148, −2.55493343355214166295021067702, −2.37868514577669762177227038126, −1.78930735016282257625297775982, −0.74264998369890049481600485601, −0.21136481435607402780141540972, 0.21136481435607402780141540972, 0.74264998369890049481600485601, 1.78930735016282257625297775982, 2.37868514577669762177227038126, 2.55493343355214166295021067702, 3.04726820625125862859212933148, 3.90565565502326318883555417914, 3.90734779431379043345770183706, 3.91829740367064132770064468083, 4.00228210919911626960026921576, 4.30837356157822123939895533126, 4.57075871711358165675726411790, 5.12846248971825754738029212440, 5.43272995429474713041559300793, 5.43810859881776782190466656253, 5.95292952459366795557344906314, 6.47276140413716617276303367793, 6.55394008362429058527231465249, 6.94772658743868467461326405431, 7.23217428890444396756726139354, 7.26878216852945703828624592089, 7.28121656645077669305103295886, 8.126851331913683768832510524418, 8.130256828769222925289323381373, 8.370542814523166876391236218295

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