Properties

Degree $5$
Conductor $20627$
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  − 2-s − 5-s − 7-s + 10-s − 11-s + 14-s + 17-s − 19-s + 22-s + 2·23-s − 29-s − 34-s + 35-s − 37-s + 38-s + 2·41-s − 2·46-s + 49-s − 53-s + 55-s + 58-s + 59-s − 61-s + 64-s − 70-s + 74-s + 77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20627 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

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

Particular Values

\[L(1/2,\rho) \approx 0.1053941200\] \[L(1,\rho) \approx 0.3369608619\]

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