Properties

Label 8-280e4-1.1-c0e4-0-0
Degree $8$
Conductor $6146560000$
Sign $1$
Analytic cond. $0.000381294$
Root an. cond. $0.373815$
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  − 2·4-s + 3·16-s − 2·49-s − 4·64-s − 2·81-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 + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 2·49-s − 4·64-s − 2·81-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 + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.000381294\)
Root analytic conductor: \(0.373815\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2222061295\)
\(L(\frac12)\) \(\approx\) \(0.2222061295\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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

−8.731615248253548933574620284630, −8.727191755292717787655958912301, −8.675369046753481445219357500126, −8.245557687611407096365883087303, −7.73410312088666485613325094090, −7.70262059668064359062657271905, −7.54135244622423929370694674942, −7.25753457813564924206943884245, −6.66647484461294080438301284079, −6.41855386499090721004024537448, −6.38754131075484860594574741791, −5.72664450290764753759245159447, −5.70606865385969620320556948495, −5.40567304635705398602777906994, −4.95575255671963414409236138270, −4.83123021718559643282327601472, −4.44019658256500854570704243426, −4.34726461970485868771277807788, −3.83008646919882056223614640972, −3.53872943663913842518785477498, −3.31518019814357706344161295430, −2.87554991319898098850264247325, −2.39305163480691963301849715955, −1.67322738982507795003024301670, −1.19131665193209874420607157325, 1.19131665193209874420607157325, 1.67322738982507795003024301670, 2.39305163480691963301849715955, 2.87554991319898098850264247325, 3.31518019814357706344161295430, 3.53872943663913842518785477498, 3.83008646919882056223614640972, 4.34726461970485868771277807788, 4.44019658256500854570704243426, 4.83123021718559643282327601472, 4.95575255671963414409236138270, 5.40567304635705398602777906994, 5.70606865385969620320556948495, 5.72664450290764753759245159447, 6.38754131075484860594574741791, 6.41855386499090721004024537448, 6.66647484461294080438301284079, 7.25753457813564924206943884245, 7.54135244622423929370694674942, 7.70262059668064359062657271905, 7.73410312088666485613325094090, 8.245557687611407096365883087303, 8.675369046753481445219357500126, 8.727191755292717787655958912301, 8.731615248253548933574620284630

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