Properties

Label 8-30e4-1.1-c2e4-0-0
Degree $8$
Conductor $810000$
Sign $1$
Analytic cond. $0.446503$
Root an. cond. $0.904124$
Motivic weight $2$
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  + 4·4-s − 16·9-s + 12·16-s + 48·19-s − 18·25-s − 128·31-s − 64·36-s + 128·49-s − 64·61-s + 32·64-s + 192·76-s − 288·79-s + 175·81-s − 72·100-s + 320·109-s − 60·121-s − 512·124-s + 127-s + 131-s + 137-s + 139-s − 192·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 4-s − 1.77·9-s + 3/4·16-s + 2.52·19-s − 0.719·25-s − 4.12·31-s − 1.77·36-s + 2.61·49-s − 1.04·61-s + 1/2·64-s + 2.52·76-s − 3.64·79-s + 2.16·81-s − 0.719·100-s + 2.93·109-s − 0.495·121-s − 4.12·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 4/3·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.446503\)
Root analytic conductor: \(0.904124\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 810000,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9970523424\)
\(L(\frac12)\) \(\approx\) \(0.9970523424\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3$C_2^2$ \( 1 + 16 T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 + 18 T^{2} + p^{4} T^{4} \)
good7$C_2^2$ \( ( 1 - 64 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 30 T^{2} + p^{4} T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
17$C_2^2$ \( ( 1 + 450 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 12 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 480 T^{2} + p^{4} T^{4} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
31$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2194 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 30 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2032 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 3168 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6690 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 8944 T^{2} + p^{4} T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2^2$ \( ( 1 + 2942 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 72 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 11856 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 11490 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 7838 T^{2} + p^{4} T^{4} )^{2} \)
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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

−12.61695966535513779904971223129, −12.00534136454441598947025714196, −11.93040687964868699036428154939, −11.71983661285254769934238067595, −11.29219669291262299039085179348, −11.06588862538936461485155837294, −10.67299930059362654735356988374, −10.46766624811856074030811447758, −9.823221613477940734412121187358, −9.415512359695815759757034805551, −9.238455499738404659373897359660, −8.746497862591469473118307090918, −8.466402456698073833816059007423, −7.83056174676368537277650251743, −7.39077479338772718174878511219, −7.28251789770048511945899769940, −6.95497593315290938568200687329, −5.93398024029878593979916958818, −5.69296378855426107335447294381, −5.67869956668041248586078808145, −5.06168367233999760498139805777, −4.01457074796258314465551376197, −3.33958289693958589849844513435, −2.97760111814052126250815104749, −1.95929403246052230983846749011, 1.95929403246052230983846749011, 2.97760111814052126250815104749, 3.33958289693958589849844513435, 4.01457074796258314465551376197, 5.06168367233999760498139805777, 5.67869956668041248586078808145, 5.69296378855426107335447294381, 5.93398024029878593979916958818, 6.95497593315290938568200687329, 7.28251789770048511945899769940, 7.39077479338772718174878511219, 7.83056174676368537277650251743, 8.466402456698073833816059007423, 8.746497862591469473118307090918, 9.238455499738404659373897359660, 9.415512359695815759757034805551, 9.823221613477940734412121187358, 10.46766624811856074030811447758, 10.67299930059362654735356988374, 11.06588862538936461485155837294, 11.29219669291262299039085179348, 11.71983661285254769934238067595, 11.93040687964868699036428154939, 12.00534136454441598947025714196, 12.61695966535513779904971223129

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