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Properties

Label 8-3360e4-1.1-c0e4-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(5.276471505\times10^{-5}\)
\(L(\frac12)\) \(\approx\) \(5.276471505\times10^{-5}\)
\(L(1)\) 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

−6.39087913118794477646735364616, −6.05315009483171128782277619590, −5.83819493397912759768301462543, −5.68634900349519436028307880440, −5.50962031112368883129233106318, −5.33879603806747154860377589541, −5.25295579043266100281683549843, −4.81486324308612423484243791029, −4.53795871700192476365173756194, −4.46514926603083440223673704098, −4.38754643103735769130362327864, −4.00008018672018261251645179101, −3.71629049351568692615436301164, −3.60120471385278982954229769164, −3.45182512484228446705640322786, −3.17852555076151465994330698562, −3.12909581580376718754805449807, −2.42721770012400945830104067697, −2.19899160916245033133772593712, −2.15857292081030146226679211993, −2.14244422097784531955252992557, −1.58184075994227475910669653191, −1.28212902969151264309184622841, −1.14504353466881640312351660644, −0.00260592241334527775899531262, 0.00260592241334527775899531262, 1.14504353466881640312351660644, 1.28212902969151264309184622841, 1.58184075994227475910669653191, 2.14244422097784531955252992557, 2.15857292081030146226679211993, 2.19899160916245033133772593712, 2.42721770012400945830104067697, 3.12909581580376718754805449807, 3.17852555076151465994330698562, 3.45182512484228446705640322786, 3.60120471385278982954229769164, 3.71629049351568692615436301164, 4.00008018672018261251645179101, 4.38754643103735769130362327864, 4.46514926603083440223673704098, 4.53795871700192476365173756194, 4.81486324308612423484243791029, 5.25295579043266100281683549843, 5.33879603806747154860377589541, 5.50962031112368883129233106318, 5.68634900349519436028307880440, 5.83819493397912759768301462543, 6.05315009483171128782277619590, 6.39087913118794477646735364616

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