Properties

Degree 4
Conductor 2617
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 2-s − 3-s + 6-s − 11-s − 13-s + 17-s + 22-s − 23-s + 26-s + 2·31-s + 32-s + 33-s − 34-s − 37-s + 39-s + 46-s − 51-s + 2·53-s + 59-s − 61-s − 2·62-s − 64-s − 66-s − 67-s + 69-s − 73-s + 74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2617 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2617\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((4,\ 2617,\ (0, 0, 1, 1:\ ),\ 1)\)

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.09639581792\] \[L(1,\rho) \approx 0.3044134854\]

Imaginary part of the first few zeros on the critical line

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