Properties

Label 8-770e4-1.1-c1e4-0-1
Degree $8$
Conductor $351530410000$
Sign $1$
Analytic cond. $1429.12$
Root an. cond. $2.47961$
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  − 8·5-s − 12·11-s − 16-s − 8·23-s + 38·25-s + 24·37-s + 28·47-s − 16·53-s + 96·55-s − 12·67-s − 32·71-s + 8·80-s − 18·81-s + 24·97-s + 12·103-s − 36·113-s + 64·115-s + 86·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 3.57·5-s − 3.61·11-s − 1/4·16-s − 1.66·23-s + 38/5·25-s + 3.94·37-s + 4.08·47-s − 2.19·53-s + 12.9·55-s − 1.46·67-s − 3.79·71-s + 0.894·80-s − 2·81-s + 2.43·97-s + 1.18·103-s − 3.38·113-s + 5.96·115-s + 7.81·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1429.12\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{770} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04714219298\)
\(L(\frac12)\) \(\approx\) \(0.04714219298\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{4} \)
5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
13$C_2^3$ \( 1 + 146 T^{4} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 3214 T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 96 T^{2} + p^{2} T^{4} )( 1 + 96 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 13294 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.43897779080180237454815393021, −7.36111786212466359651046600034, −7.32285069290052777824897882601, −6.93426699676010219117513401201, −6.57874254139703300076035958178, −6.11739477291568467300617191012, −5.92360388728646491487265115320, −5.75330273446599044718304449550, −5.70878160755590499972270568385, −5.08163076776503735119210157954, −5.01294941567582544771760225709, −4.44463948372988282171460184951, −4.38540375158162429065075813470, −4.35821044721616690263370439198, −4.31939352958825311363591430287, −3.70040971250586905111789545760, −3.44797687424685792106139419462, −3.08699148705917981377178459384, −2.86556166036080567521721632173, −2.62603718125499673843228938150, −2.50947526449916942830668017152, −1.97923180520971605931022434698, −1.13953218114641707511722086791, −0.65530401066015364356566676435, −0.096926374457960697344707824981, 0.096926374457960697344707824981, 0.65530401066015364356566676435, 1.13953218114641707511722086791, 1.97923180520971605931022434698, 2.50947526449916942830668017152, 2.62603718125499673843228938150, 2.86556166036080567521721632173, 3.08699148705917981377178459384, 3.44797687424685792106139419462, 3.70040971250586905111789545760, 4.31939352958825311363591430287, 4.35821044721616690263370439198, 4.38540375158162429065075813470, 4.44463948372988282171460184951, 5.01294941567582544771760225709, 5.08163076776503735119210157954, 5.70878160755590499972270568385, 5.75330273446599044718304449550, 5.92360388728646491487265115320, 6.11739477291568467300617191012, 6.57874254139703300076035958178, 6.93426699676010219117513401201, 7.32285069290052777824897882601, 7.36111786212466359651046600034, 7.43897779080180237454815393021

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