Properties

Label 8-2160e4-1.1-c1e4-0-6
Degree $8$
Conductor $2.177\times 10^{13}$
Sign $1$
Analytic cond. $88495.9$
Root an. cond. $4.15303$
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  + 4·19-s + 8·25-s − 8·31-s + 10·49-s − 52·61-s − 20·79-s + 32·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 0.917·19-s + 8/5·25-s − 1.43·31-s + 10/7·49-s − 6.65·61-s − 2.25·79-s + 3.06·109-s − 0.727·121-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 + 2.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(88495.9\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.055140711\)
\(L(\frac12)\) \(\approx\) \(1.055140711\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
good7$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 185 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.48763420572749446520304207625, −6.26801554792319757406757778921, −5.92298764961018639483372774224, −5.81392971546344827669767195182, −5.65111146848112257354826223966, −5.54230579613757813545824362147, −5.14083940833835182768076769999, −4.85884780578900376375661737750, −4.82280637084575706822080377461, −4.51518465739885644886076951116, −4.38204568320391849710925709148, −4.08921211569861187159800963719, −3.97788520306310568290775809653, −3.38602517921363753763135758346, −3.34312120071628030093462920203, −3.10262559583268932757745261390, −2.88746944721734086384559864569, −2.87910346371919771687798427180, −2.28117007132935914284038724616, −1.98564635760011390066467398133, −1.76002275331648304430222086535, −1.49420487921238821708445929026, −1.08510187678724062339329611421, −0.832724645568873429735610497587, −0.18121793090740487203286781442, 0.18121793090740487203286781442, 0.832724645568873429735610497587, 1.08510187678724062339329611421, 1.49420487921238821708445929026, 1.76002275331648304430222086535, 1.98564635760011390066467398133, 2.28117007132935914284038724616, 2.87910346371919771687798427180, 2.88746944721734086384559864569, 3.10262559583268932757745261390, 3.34312120071628030093462920203, 3.38602517921363753763135758346, 3.97788520306310568290775809653, 4.08921211569861187159800963719, 4.38204568320391849710925709148, 4.51518465739885644886076951116, 4.82280637084575706822080377461, 4.85884780578900376375661737750, 5.14083940833835182768076769999, 5.54230579613757813545824362147, 5.65111146848112257354826223966, 5.81392971546344827669767195182, 5.92298764961018639483372774224, 6.26801554792319757406757778921, 6.48763420572749446520304207625

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