Properties

Label 8-252e4-1.1-c1e4-0-13
Degree $8$
Conductor $4032758016$
Sign $1$
Analytic cond. $16.3949$
Root an. cond. $1.41853$
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 + 6·5-s + 4·8-s + 6·9-s + 12·10-s − 6·13-s + 8·16-s + 12·18-s + 12·20-s + 11·25-s − 12·26-s − 10·29-s + 8·32-s + 12·36-s + 24·40-s − 30·41-s + 36·45-s − 2·49-s + 22·50-s − 12·52-s + 16·53-s − 20·58-s + 18·61-s + 8·64-s − 36·65-s + 24·72-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 2.68·5-s + 1.41·8-s + 2·9-s + 3.79·10-s − 1.66·13-s + 2·16-s + 2.82·18-s + 2.68·20-s + 11/5·25-s − 2.35·26-s − 1.85·29-s + 1.41·32-s + 2·36-s + 3.79·40-s − 4.68·41-s + 5.36·45-s − 2/7·49-s + 3.11·50-s − 1.66·52-s + 2.19·53-s − 2.62·58-s + 2.30·61-s + 64-s − 4.46·65-s + 2.82·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(16.3949\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.614378157\)
\(L(\frac12)\) \(\approx\) \(7.614378157\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good5$C_2^2$ \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 21 T^{2} - 88 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$$\times$$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 59 T^{2} + p^{2} T^{4} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 67 T^{2} + 2280 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
59$C_2^3$ \( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 91 T^{2} + 3792 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^3$ \( 1 + 149 T^{2} + 15960 T^{4} + 149 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^3$ \( 1 - 163 T^{2} + 19680 T^{4} - 163 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
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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

−8.976855697776746668278763834093, −8.506660253201418285764253109204, −8.214232252471624490554566547961, −7.960929673871615105542199046822, −7.36551539489048063124601893886, −7.35983071920907917120746718290, −7.27993512837188473218176482476, −6.69402091778525476337320543773, −6.66933734433112663769341375752, −6.45462138982227393152421919620, −6.00379419664960158252055012288, −5.53566865922893692989759711715, −5.37282048949243517978431143646, −5.31908346654087518336461275528, −5.00909661241390223590312293124, −4.78691729555622263933618974886, −4.31715631829141261381732144300, −3.84598860026730180288414799812, −3.84068279230267176640335174045, −3.41211065654564407845841869545, −2.71876645157251127892315917193, −2.28882399713801770003505012401, −1.89256466737531534176309879737, −1.81910589390400723078534820896, −1.38502692382542142657589191908, 1.38502692382542142657589191908, 1.81910589390400723078534820896, 1.89256466737531534176309879737, 2.28882399713801770003505012401, 2.71876645157251127892315917193, 3.41211065654564407845841869545, 3.84068279230267176640335174045, 3.84598860026730180288414799812, 4.31715631829141261381732144300, 4.78691729555622263933618974886, 5.00909661241390223590312293124, 5.31908346654087518336461275528, 5.37282048949243517978431143646, 5.53566865922893692989759711715, 6.00379419664960158252055012288, 6.45462138982227393152421919620, 6.66933734433112663769341375752, 6.69402091778525476337320543773, 7.27993512837188473218176482476, 7.35983071920907917120746718290, 7.36551539489048063124601893886, 7.960929673871615105542199046822, 8.214232252471624490554566547961, 8.506660253201418285764253109204, 8.976855697776746668278763834093

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