Properties

Degree $2$
Conductor $639$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

Learn more

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 1.80·2-s + 2.24·4-s − 1.24·5-s + 2.24·8-s − 2.24·10-s + 1.80·16-s − 1.80·19-s − 2.80·20-s + 0.554·25-s + 0.445·29-s + 1.00·32-s − 1.80·37-s − 3.24·38-s − 2.80·40-s + 1.24·43-s + 49-s + 1.00·50-s + 0.801·58-s − 71-s + 1.24·73-s − 3.24·74-s − 4.04·76-s + 1.24·79-s − 2.24·80-s + 1.80·83-s + 2.24·86-s + 0.445·89-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

\[L(1/2,\rho) \approx 2.039039874\] \[L(1,\rho) \approx 2.088546630\]

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