Properties

Degree $5$
Conductor $425473129$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 4-s − 7-s − 8-s + 2·11-s + 16-s − 17-s − 19-s − 23-s + 25-s − 28-s − 32-s − 41-s + 2·44-s + 49-s − 53-s + 56-s − 59-s − 61-s + 2·64-s − 68-s + 71-s − 76-s − 2·77-s − 83-s − 2·88-s + 2·89-s − 92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(20627^{2}\right)^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{4} \, L(s,\rho)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(5\)
Conductor: \(20627^{2}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((5,\ 20627^{2} ,\ ( 0, 1, 1, 1, 1 : \ ),\ 1 )\)

Particular Values

\[L(1/2,\rho) \approx 1.685463457\] \[L(1,\rho) \approx 1.073472984\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{5} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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