Properties

Label 8-1400e4-1.1-c0e4-0-5
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $0.238309$
Root an. cond. $0.835877$
Motivic weight $0$
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 + 4·9-s + 4·36-s − 2·49-s − 64-s + 4·71-s − 4·79-s + 10·81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + ⋯
L(s)  = 1  + 4-s + 4·9-s + 4·36-s − 2·49-s − 64-s + 4·71-s − 4·79-s + 10·81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.238309\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.219330537\)
\(L(\frac12)\) \(\approx\) \(2.219330537\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_2$ \( ( 1 - T + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T + T^{2} )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.95326472785480355241422688688, −6.86453488460322438690075566461, −6.70527757626231824758033501349, −6.50277314686949515194335874459, −6.44221876745650083029023119916, −5.95825078169335581832940843414, −5.76213838251622997730402030452, −5.57637640297452060270368011031, −5.23719777479881426953485330547, −4.85128989002180292285306507561, −4.68097393040551552482862211262, −4.65668643985632725499620215615, −4.44715184985955166488361882859, −4.11814511370719721310756726033, −3.78360694597269156595348377952, −3.67399747289231948260723593332, −3.33390983473758282201081445631, −3.32281008630357227150847568278, −2.49039640507032635732273873908, −2.42266313709867637572671166885, −2.31106944723953951983158795480, −1.77614059708378109395401590793, −1.52034457622724447022130108464, −1.23622711660360511502588434856, −1.13550396773616017793949129346, 1.13550396773616017793949129346, 1.23622711660360511502588434856, 1.52034457622724447022130108464, 1.77614059708378109395401590793, 2.31106944723953951983158795480, 2.42266313709867637572671166885, 2.49039640507032635732273873908, 3.32281008630357227150847568278, 3.33390983473758282201081445631, 3.67399747289231948260723593332, 3.78360694597269156595348377952, 4.11814511370719721310756726033, 4.44715184985955166488361882859, 4.65668643985632725499620215615, 4.68097393040551552482862211262, 4.85128989002180292285306507561, 5.23719777479881426953485330547, 5.57637640297452060270368011031, 5.76213838251622997730402030452, 5.95825078169335581832940843414, 6.44221876745650083029023119916, 6.50277314686949515194335874459, 6.70527757626231824758033501349, 6.86453488460322438690075566461, 6.95326472785480355241422688688

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