Properties

Label 8-320e4-1.1-c1e4-0-3
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $42.6293$
Root an. cond. $1.59850$
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  + 4·5-s − 4·13-s + 4·17-s + 2·25-s − 20·37-s + 8·41-s + 28·53-s − 24·61-s − 16·65-s − 28·73-s + 18·81-s + 16·85-s − 28·97-s + 40·101-s + 36·113-s + 20·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.10·13-s + 0.970·17-s + 2/5·25-s − 3.28·37-s + 1.24·41-s + 3.84·53-s − 3.07·61-s − 1.98·65-s − 3.27·73-s + 2·81-s + 1.73·85-s − 2.84·97-s + 3.98·101-s + 3.38·113-s + 1.81·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(42.6293\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.460927437\)
\(L(\frac12)\) \(\approx\) \(2.460927437\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good3$C_2$$\times$$C_2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \)
7$C_2^3$ \( 1 - 34 T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 542 T^{4} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
43$C_2^3$ \( 1 + 2702 T^{4} + p^{4} T^{8} \)
47$C_2^3$ \( 1 + 3326 T^{4} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 - 2578 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^3$ \( 1 + 2606 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.520030397748944726355109838589, −8.242490332142431549097990350949, −7.905255167452555018665217257026, −7.66452995186847811374692307695, −7.40134027715321106397179914974, −7.06817466296699457556799177892, −6.99415616841921703025163970581, −6.82308850182998501342410649284, −6.20733675697868705625716078948, −6.00283143055561027617739772875, −5.85169404000169724182152534028, −5.51534163991162366790102186721, −5.44971124627777014870701033416, −5.20647333192485474939450409220, −4.56981473109667500251945808067, −4.39256177429860271760739604785, −4.37068983109001058438863996460, −3.51405606391579631897399383262, −3.31913861834550304979634410517, −3.17273449765967984060066023065, −2.46614160490971712677917911034, −2.27341516007885997535894760599, −1.75002397272155427007061779895, −1.66652469312998831091684148606, −0.67909613830254790842068001412, 0.67909613830254790842068001412, 1.66652469312998831091684148606, 1.75002397272155427007061779895, 2.27341516007885997535894760599, 2.46614160490971712677917911034, 3.17273449765967984060066023065, 3.31913861834550304979634410517, 3.51405606391579631897399383262, 4.37068983109001058438863996460, 4.39256177429860271760739604785, 4.56981473109667500251945808067, 5.20647333192485474939450409220, 5.44971124627777014870701033416, 5.51534163991162366790102186721, 5.85169404000169724182152534028, 6.00283143055561027617739772875, 6.20733675697868705625716078948, 6.82308850182998501342410649284, 6.99415616841921703025163970581, 7.06817466296699457556799177892, 7.40134027715321106397179914974, 7.66452995186847811374692307695, 7.905255167452555018665217257026, 8.242490332142431549097990350949, 8.520030397748944726355109838589

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