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Properties

Label 8-2340e4-1.1-c0e4-0-12
Degree $8$
Conductor $2.998\times 10^{13}$
Sign $1$
Analytic cond. $1.85990$
Root an. cond. $1.08065$
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  + 4·13-s − 16-s + 4·37-s + 8·61-s − 4·73-s − 4·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4·13-s − 16-s + 4·37-s + 8·61-s − 4·73-s − 4·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.85990\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.142428209\)
\(L(\frac12)\) \(\approx\) \(2.142428209\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3 \( 1 \)
5$C_2^2$ \( 1 + T^{4} \)
13$C_1$ \( ( 1 - T )^{4} \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$ \( ( 1 - T )^{8} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
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_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
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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

−6.64747098805120691184909267847, −6.35175253581084763014283145672, −6.08668790884684171125330374672, −6.05859654140790274552649071012, −5.70452837210724423017190116379, −5.58276539574117155910167620115, −5.54317726806590554613646252838, −5.17391405350923039570019567132, −5.00238914735269829362090752738, −4.51281808394313480380039980726, −4.37847692682159068456559231276, −4.04181494028507596218540911966, −4.01206291634033377387879825591, −3.94146379536193648027774343567, −3.77404026685464120427890007793, −3.24465530149377068280897280932, −3.16469496419755024217743693587, −2.75881612795767472752281658457, −2.49571025435960771538350945956, −2.42781610075457734949426527156, −2.06339675318706831918557671464, −1.54593821261901323970231263088, −1.28672435472889844620330286073, −1.02461204492146758925674189109, −0.869590929755249138369379602033, 0.869590929755249138369379602033, 1.02461204492146758925674189109, 1.28672435472889844620330286073, 1.54593821261901323970231263088, 2.06339675318706831918557671464, 2.42781610075457734949426527156, 2.49571025435960771538350945956, 2.75881612795767472752281658457, 3.16469496419755024217743693587, 3.24465530149377068280897280932, 3.77404026685464120427890007793, 3.94146379536193648027774343567, 4.01206291634033377387879825591, 4.04181494028507596218540911966, 4.37847692682159068456559231276, 4.51281808394313480380039980726, 5.00238914735269829362090752738, 5.17391405350923039570019567132, 5.54317726806590554613646252838, 5.58276539574117155910167620115, 5.70452837210724423017190116379, 6.05859654140790274552649071012, 6.08668790884684171125330374672, 6.35175253581084763014283145672, 6.64747098805120691184909267847

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