Properties

Degree $2$
Conductor $639$
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  + 0.445·2-s − 0.801·4-s + 1.80·5-s − 0.801·8-s + 0.801·10-s + 0.445·16-s − 0.445·19-s − 1.44·20-s + 2.24·25-s − 1.24·29-s + 1.00·32-s − 0.445·37-s − 0.198·38-s − 1.44·40-s − 1.80·43-s + 49-s + 1.00·50-s − 0.554·58-s − 71-s − 1.80·73-s − 0.198·74-s + 0.356·76-s − 1.80·79-s + 0.801·80-s + 0.445·83-s − 0.801·86-s − 1.24·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 639,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 1.207028707\] \[L(1,\rho) \approx 1.278301687\]

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