Properties

Degree $2$
Conductor $562636$
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  + 2·5-s + 9-s − 11-s − 13-s − 17-s − 19-s + 2·23-s + 3·25-s − 29-s − 31-s + 2·45-s − 47-s + 49-s − 2·55-s − 2·65-s − 73-s − 79-s + 81-s − 83-s − 2·85-s − 89-s − 2·95-s − 97-s − 99-s − 109-s − 113-s + 4·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 562636 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(562636\)    =    \(2^{2} \cdot 140659\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 562636,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 2.779612436\] \[L(1,\rho) \approx 1.374953972\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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