Properties

Label 8-630e4-1.1-c1e4-0-7
Degree $8$
Conductor $157529610000$
Sign $1$
Analytic cond. $640.428$
Root an. cond. $2.24289$
Motivic weight $1$
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  + 4-s − 4·5-s + 2·7-s − 3·9-s + 12·13-s + 6·19-s − 4·20-s + 10·25-s + 2·28-s − 12·29-s − 18·31-s − 8·35-s − 3·36-s − 2·37-s − 18·41-s + 4·43-s + 12·45-s − 18·47-s − 11·49-s + 12·52-s + 6·53-s − 6·59-s + 24·61-s − 6·63-s − 64-s − 48·65-s + 8·67-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s + 0.755·7-s − 9-s + 3.32·13-s + 1.37·19-s − 0.894·20-s + 2·25-s + 0.377·28-s − 2.22·29-s − 3.23·31-s − 1.35·35-s − 1/2·36-s − 0.328·37-s − 2.81·41-s + 0.609·43-s + 1.78·45-s − 2.62·47-s − 1.57·49-s + 1.66·52-s + 0.824·53-s − 0.781·59-s + 3.07·61-s − 0.755·63-s − 1/8·64-s − 5.95·65-s + 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(640.428\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.303511684\)
\(L(\frac12)\) \(\approx\) \(1.303511684\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
good11$D_4\times C_2$ \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 4 T^{2} - 666 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 12 T + 109 T^{2} + 732 T^{3} + 4272 T^{4} + 732 p T^{5} + 109 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 18 T + 188 T^{2} + 1440 T^{3} + 8787 T^{4} + 1440 p T^{5} + 188 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 2 T - 44 T^{2} - 52 T^{3} + 787 T^{4} - 52 p T^{5} - 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 4 T - 47 T^{2} + 92 T^{3} + 1432 T^{4} + 92 p T^{5} - 47 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 6 T + 40 T^{2} - 168 T^{3} - 1389 T^{4} - 168 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 108 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 24 T + 326 T^{2} - 3216 T^{3} + 25947 T^{4} - 3216 p T^{5} + 326 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 260 T^{2} + 26874 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 2 T + 88 T^{2} + 484 T^{3} + 499 T^{4} + 484 p T^{5} + 88 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 12 T - 31 T^{2} - 108 T^{3} + 11784 T^{4} - 108 p T^{5} - 31 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 12 T + 38 T^{2} + 864 T^{3} - 9501 T^{4} + 864 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 12 T + 110 T^{2} + 744 T^{3} - 909 T^{4} + 744 p T^{5} + 110 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.79607652137264196970515844042, −7.35421664463322503674973931848, −7.30061577312048384882896132546, −6.80679567220131257097164024274, −6.68928564122393109300222088875, −6.65907050825912759500501845490, −6.29352844880963334019350224388, −5.90931070942197429662999659403, −5.64435730066853738584944188419, −5.29368411791166463522667467373, −5.23725999504448322484085722535, −5.21127362825560431148331972097, −4.79488583006067507892290530226, −4.17227448173198733089592572134, −3.90010254820835112747678771613, −3.80349759170134503101220163200, −3.52683880503321538642398586293, −3.37429495985151954643688675634, −3.29199062693896075769752840866, −2.81480228706294801346306141533, −1.96249970031881433156276035156, −1.94552673553787370584648428394, −1.56261958949668680484042128166, −1.08682525589808161607387736132, −0.35889196522443064424264991280, 0.35889196522443064424264991280, 1.08682525589808161607387736132, 1.56261958949668680484042128166, 1.94552673553787370584648428394, 1.96249970031881433156276035156, 2.81480228706294801346306141533, 3.29199062693896075769752840866, 3.37429495985151954643688675634, 3.52683880503321538642398586293, 3.80349759170134503101220163200, 3.90010254820835112747678771613, 4.17227448173198733089592572134, 4.79488583006067507892290530226, 5.21127362825560431148331972097, 5.23725999504448322484085722535, 5.29368411791166463522667467373, 5.64435730066853738584944188419, 5.90931070942197429662999659403, 6.29352844880963334019350224388, 6.65907050825912759500501845490, 6.68928564122393109300222088875, 6.80679567220131257097164024274, 7.30061577312048384882896132546, 7.35421664463322503674973931848, 7.79607652137264196970515844042

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