Properties

Label 8-244e4-1.1-c0e4-0-0
Degree $8$
Conductor $3544535296$
Sign $1$
Analytic cond. $0.000219881$
Root an. cond. $0.348958$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·5-s − 9-s − 3·10-s − 2·13-s − 2·17-s + 18-s + 6·25-s + 2·26-s − 2·29-s + 32-s + 2·34-s − 2·37-s − 2·41-s − 3·45-s − 49-s − 6·50-s + 3·53-s + 2·58-s − 61-s − 64-s − 6·65-s − 2·73-s + 2·74-s + 2·82-s − 6·85-s + 3·89-s + ⋯
L(s)  = 1  − 2-s + 3·5-s − 9-s − 3·10-s − 2·13-s − 2·17-s + 18-s + 6·25-s + 2·26-s − 2·29-s + 32-s + 2·34-s − 2·37-s − 2·41-s − 3·45-s − 49-s − 6·50-s + 3·53-s + 2·58-s − 61-s − 64-s − 6·65-s − 2·73-s + 2·74-s + 2·82-s − 6·85-s + 3·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 61^{4}\)
Sign: $1$
Analytic conductor: \(0.000219881\)
Root analytic conductor: \(0.348958\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 61^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2133322465\)
\(L(\frac12)\) \(\approx\) \(0.2133322465\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
61$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
5$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−9.103478022571611103698751662218, −8.732058650871246485015262464303, −8.684704932872452075491222471335, −8.606203066781969818400924984915, −8.549347688283605609729142946833, −7.52202050583379250182937713569, −7.47111032825195387802704109835, −7.43672014559604176354689487191, −6.91410371957520061958330593574, −6.65723171380626883149697443706, −6.37443483358141840272337402600, −6.20536594115554305913447657607, −5.92043287394811455548647486446, −5.48251791481274009607292464551, −5.34800628034127901105470219065, −4.92728100258333235645881544918, −4.87925060063891447259781380224, −4.60106541389467582366599801433, −3.91980923375694544646322057930, −3.34431378684299440643763237606, −3.08670258509988933287752498379, −2.47511045532847256016510563952, −2.26072258942276964586639132378, −2.04615428070157945945136219029, −1.58874477707428449180824401754, 1.58874477707428449180824401754, 2.04615428070157945945136219029, 2.26072258942276964586639132378, 2.47511045532847256016510563952, 3.08670258509988933287752498379, 3.34431378684299440643763237606, 3.91980923375694544646322057930, 4.60106541389467582366599801433, 4.87925060063891447259781380224, 4.92728100258333235645881544918, 5.34800628034127901105470219065, 5.48251791481274009607292464551, 5.92043287394811455548647486446, 6.20536594115554305913447657607, 6.37443483358141840272337402600, 6.65723171380626883149697443706, 6.91410371957520061958330593574, 7.43672014559604176354689487191, 7.47111032825195387802704109835, 7.52202050583379250182937713569, 8.549347688283605609729142946833, 8.606203066781969818400924984915, 8.684704932872452075491222471335, 8.732058650871246485015262464303, 9.103478022571611103698751662218

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