Properties

Label 8-260e4-1.1-c1e4-0-4
Degree $8$
Conductor $4569760000$
Sign $1$
Analytic cond. $18.5781$
Root an. cond. $1.44087$
Motivic weight $1$
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  − 2·2-s + 2·4-s + 4·5-s − 4·8-s − 8·10-s − 4·13-s + 8·16-s + 14·17-s + 8·20-s + 5·25-s + 8·26-s + 30·29-s − 8·32-s − 28·34-s − 8·37-s − 16·40-s + 24·41-s − 10·50-s − 8·52-s − 10·53-s − 60·58-s − 12·61-s + 8·64-s − 16·65-s + 28·68-s − 22·73-s + 16·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s − 1.41·8-s − 2.52·10-s − 1.10·13-s + 2·16-s + 3.39·17-s + 1.78·20-s + 25-s + 1.56·26-s + 5.57·29-s − 1.41·32-s − 4.80·34-s − 1.31·37-s − 2.52·40-s + 3.74·41-s − 1.41·50-s − 1.10·52-s − 1.37·53-s − 7.87·58-s − 1.53·61-s + 64-s − 1.98·65-s + 3.39·68-s − 2.57·73-s + 1.85·74-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(18.5781\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{260} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.300859861\)
\(L(\frac12)\) \(\approx\) \(1.300859861\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
good3$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
7$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$$\times$$C_2^2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
41$C_2$$\times$$C_2^2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \)
43$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
59$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$$\times$$C_2^2$ \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \)
67$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
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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

−8.772771927299445725384838735213, −8.371000029517526354214094418073, −8.166095409120280519768180337559, −8.131400654565639397295644617651, −8.031606914787126180474967362253, −7.30584786406429692685882214829, −7.27866651381943936682065003800, −7.13353433799032811832697317839, −6.44201946370872960518714546572, −6.35180832497297740861400420421, −6.12740126535897525430798517549, −5.96617526425806339880635493458, −5.50659100733429146511757047446, −5.25933939926732076545641023941, −5.17643850111345254423070282768, −4.68089880556546600524935083596, −4.12372830199504256312742633006, −4.07024148128909104046173651961, −3.10735196321887568127731194372, −2.78129623757159189916483365992, −2.77270746314136231897448015067, −2.74673051334977702377817061566, −1.55798000021671217553507804455, −1.35343866750416102201107010287, −0.866992351343229496059516783790, 0.866992351343229496059516783790, 1.35343866750416102201107010287, 1.55798000021671217553507804455, 2.74673051334977702377817061566, 2.77270746314136231897448015067, 2.78129623757159189916483365992, 3.10735196321887568127731194372, 4.07024148128909104046173651961, 4.12372830199504256312742633006, 4.68089880556546600524935083596, 5.17643850111345254423070282768, 5.25933939926732076545641023941, 5.50659100733429146511757047446, 5.96617526425806339880635493458, 6.12740126535897525430798517549, 6.35180832497297740861400420421, 6.44201946370872960518714546572, 7.13353433799032811832697317839, 7.27866651381943936682065003800, 7.30584786406429692685882214829, 8.031606914787126180474967362253, 8.131400654565639397295644617651, 8.166095409120280519768180337559, 8.371000029517526354214094418073, 8.772771927299445725384838735213

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