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Properties

Label 8-775e4-1.1-c0e4-0-1
Degree $8$
Conductor $360750390625$
Sign $1$
Analytic cond. $0.0223787$
Root an. cond. $0.621912$
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  + 3·2-s + 6·4-s − 5-s − 2·7-s + 10·8-s − 9-s − 3·10-s − 6·14-s + 15·16-s − 3·18-s + 3·19-s − 6·20-s − 12·28-s − 31-s + 22·32-s + 2·35-s − 6·36-s + 9·38-s − 10·40-s − 2·41-s + 45-s − 2·47-s + 49-s − 20·56-s − 2·59-s − 3·62-s + 2·63-s + ⋯
L(s)  = 1  + 3·2-s + 6·4-s − 5-s − 2·7-s + 10·8-s − 9-s − 3·10-s − 6·14-s + 15·16-s − 3·18-s + 3·19-s − 6·20-s − 12·28-s − 31-s + 22·32-s + 2·35-s − 6·36-s + 9·38-s − 10·40-s − 2·41-s + 45-s − 2·47-s + 49-s − 20·56-s − 2·59-s − 3·62-s + 2·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(0.0223787\)
Root analytic conductor: \(0.621912\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 31^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.830826309\)
\(L(\frac12)\) \(\approx\) \(3.830826309\)
\(L(1)\) 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)$
bad5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
31$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good2$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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

−7.64811746251973525493732633789, −7.17024409315532894496199105191, −7.05560692864543377322886029772, −6.82506254252458910430393163903, −6.43703942607549068384431573818, −6.40017661982809519515922050432, −6.32385843868541750037774545088, −6.04983149419680350203424714745, −5.49690667836260362645455957223, −5.40526509856336253553676443423, −5.32852086698375049938421213693, −5.17254395531800566999360196404, −4.71909648391123343508513486815, −4.60532165955001197682012470155, −4.15960991153094321852977362358, −3.77529144025918400219114578536, −3.62758019476594354013153900710, −3.61419537694926660339914449926, −3.04926713031793471149493833678, −3.02073880753934556267769911662, −2.96915284520705345769319498153, −2.59646677040222676282941517460, −2.06862563894705034047022724950, −1.45820259384427566239885733439, −1.32266339271914437299760422737, 1.32266339271914437299760422737, 1.45820259384427566239885733439, 2.06862563894705034047022724950, 2.59646677040222676282941517460, 2.96915284520705345769319498153, 3.02073880753934556267769911662, 3.04926713031793471149493833678, 3.61419537694926660339914449926, 3.62758019476594354013153900710, 3.77529144025918400219114578536, 4.15960991153094321852977362358, 4.60532165955001197682012470155, 4.71909648391123343508513486815, 5.17254395531800566999360196404, 5.32852086698375049938421213693, 5.40526509856336253553676443423, 5.49690667836260362645455957223, 6.04983149419680350203424714745, 6.32385843868541750037774545088, 6.40017661982809519515922050432, 6.43703942607549068384431573818, 6.82506254252458910430393163903, 7.05560692864543377322886029772, 7.17024409315532894496199105191, 7.64811746251973525493732633789

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