Properties

Degree $8$
Conductor $430467210000$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·5-s + 4·11-s − 4·20-s + 5·25-s + 16·31-s + 4·41-s + 4·44-s − 10·49-s − 16·55-s − 20·59-s − 4·61-s − 64-s − 48·71-s − 40·89-s + 5·100-s − 16·101-s − 40·109-s + 26·121-s + 16·124-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s + 1.20·11-s − 0.894·20-s + 25-s + 2.87·31-s + 0.624·41-s + 0.603·44-s − 1.42·49-s − 2.15·55-s − 2.60·59-s − 0.512·61-s − 1/8·64-s − 5.69·71-s − 4.23·89-s + 1/2·100-s − 1.59·101-s − 3.83·109-s + 2.36·121-s + 1.43·124-s + 0.357·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{810} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.120970\)
\(L(\frac12)\) \(\approx\) \(0.120970\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3 \( 1 \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
good7$C_2^3$ \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^3$ \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^3$ \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
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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

−7.43395902468856121834926917469, −7.26943802593790705951199392955, −6.79068810139700082072572490684, −6.70927112430579125143091704597, −6.47949295941319424134878707995, −6.43385647378588617098655425467, −5.85438300087007503244714141412, −5.77469119071027812804284212739, −5.76300972825635462719417179980, −5.12939463696529160850278545433, −4.93710568150025938909243936881, −4.52330383227847479585652771621, −4.40049003068268393968876368221, −4.20432973635725800949346151524, −4.00562393910934380269582465942, −3.96175391746545540882408438466, −3.10341379116708057205929908035, −3.07480577387326517516897501884, −3.01972959333313213593525001656, −2.75971511172415920091243323121, −2.17890744469028945495890594484, −1.59464547805471948132622686554, −1.28756888547516624247362115688, −1.21192733268552206585795223702, −0.091413956349506470307452914209, 0.091413956349506470307452914209, 1.21192733268552206585795223702, 1.28756888547516624247362115688, 1.59464547805471948132622686554, 2.17890744469028945495890594484, 2.75971511172415920091243323121, 3.01972959333313213593525001656, 3.07480577387326517516897501884, 3.10341379116708057205929908035, 3.96175391746545540882408438466, 4.00562393910934380269582465942, 4.20432973635725800949346151524, 4.40049003068268393968876368221, 4.52330383227847479585652771621, 4.93710568150025938909243936881, 5.12939463696529160850278545433, 5.76300972825635462719417179980, 5.77469119071027812804284212739, 5.85438300087007503244714141412, 6.43385647378588617098655425467, 6.47949295941319424134878707995, 6.70927112430579125143091704597, 6.79068810139700082072572490684, 7.26943802593790705951199392955, 7.43395902468856121834926917469

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