Properties

Label 8-1323e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.064\times 10^{12}$
Sign $1$
Analytic cond. $12455.1$
Root an. cond. $3.25026$
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  + 4·4-s + 4·16-s − 8·25-s − 16·37-s − 28·43-s − 16·64-s + 8·67-s − 16·79-s − 32·100-s + 68·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s − 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s − 112·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·4-s + 16-s − 8/5·25-s − 2.63·37-s − 4.26·43-s − 2·64-s + 0.977·67-s − 1.80·79-s − 3.19·100-s + 6.51·109-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s − 8.53·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(12455.1\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.441726221\)
\(L(\frac12)\) \(\approx\) \(1.441726221\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 172 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.83319493251373109573172059498, −6.73160133576794666526167745560, −6.71452128124924412308042354250, −6.30059493774842912022288552537, −5.90061519650483049965513452327, −5.81106970216465401383198068022, −5.79640001796900568062568531768, −5.34762215962010658576289169963, −5.09697202808740114855547135786, −5.01825711218003543472877545258, −4.52564831901881287593505763555, −4.41702198243998936206426881347, −4.27648783720070339870586111494, −3.66054628467012157156542461863, −3.58292376666547518646566400215, −3.22793677079083761948203997340, −3.14775754718984525975416580052, −2.97158607981496238445906936250, −2.46993964789137036973202544275, −1.97323895603070491724512981918, −1.94569743290888129235309167133, −1.83388915671197582229556858611, −1.62207956988385369241043138511, −0.915507227025875453357793723236, −0.21301761024486320623067274634, 0.21301761024486320623067274634, 0.915507227025875453357793723236, 1.62207956988385369241043138511, 1.83388915671197582229556858611, 1.94569743290888129235309167133, 1.97323895603070491724512981918, 2.46993964789137036973202544275, 2.97158607981496238445906936250, 3.14775754718984525975416580052, 3.22793677079083761948203997340, 3.58292376666547518646566400215, 3.66054628467012157156542461863, 4.27648783720070339870586111494, 4.41702198243998936206426881347, 4.52564831901881287593505763555, 5.01825711218003543472877545258, 5.09697202808740114855547135786, 5.34762215962010658576289169963, 5.79640001796900568062568531768, 5.81106970216465401383198068022, 5.90061519650483049965513452327, 6.30059493774842912022288552537, 6.71452128124924412308042354250, 6.73160133576794666526167745560, 6.83319493251373109573172059498

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