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Properties

Label 8-3360e4-1.1-c0e4-0-2
Degree $8$
Conductor $1.275\times 10^{14}$
Sign $1$
Analytic cond. $7.90652$
Root an. cond. $1.29493$
Motivic weight $0$
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  + 4·23-s − 4·37-s − 81-s − 4·107-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 4·23-s − 4·37-s − 81-s − 4·107-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7.90652\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.292477575\)
\(L(\frac12)\) \(\approx\) \(1.292477575\)
\(L(1)\) 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 \( 1 \)
3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
good11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.37698000436167948237482238110, −6.11647133080889287714858957776, −5.65541706252530411636655841555, −5.62981403016433621684020422534, −5.61460365937993967388164914666, −5.12502765818217918523432446890, −5.03781726724227775418070512940, −5.03411739495281885285845685383, −4.77633598689207806039503849643, −4.63581582410514797538010248763, −4.21081918363286643490336387099, −3.88846440596530211642405574198, −3.72948967295506741411680163173, −3.72857078874404254294913784706, −3.43187919129754962136777984161, −2.99367160584725233435958537431, −2.78534157540500777561613026980, −2.78498810814441376137336085160, −2.68033255828806444843775289344, −2.10582144829845530470220500673, −1.72102022850308997771877137893, −1.62325491945416098093834723918, −1.20395491585040030172455819834, −1.16295158285315436460594691225, −0.44389406066761480172771924739, 0.44389406066761480172771924739, 1.16295158285315436460594691225, 1.20395491585040030172455819834, 1.62325491945416098093834723918, 1.72102022850308997771877137893, 2.10582144829845530470220500673, 2.68033255828806444843775289344, 2.78498810814441376137336085160, 2.78534157540500777561613026980, 2.99367160584725233435958537431, 3.43187919129754962136777984161, 3.72857078874404254294913784706, 3.72948967295506741411680163173, 3.88846440596530211642405574198, 4.21081918363286643490336387099, 4.63581582410514797538010248763, 4.77633598689207806039503849643, 5.03411739495281885285845685383, 5.03781726724227775418070512940, 5.12502765818217918523432446890, 5.61460365937993967388164914666, 5.62981403016433621684020422534, 5.65541706252530411636655841555, 6.11647133080889287714858957776, 6.37698000436167948237482238110

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