Properties

Degree $2$
Conductor $3639$
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  + 1.61·2-s + 3-s + 1.61·4-s + 0.209·5-s + 1.61·6-s − 0.209·7-s + 0.999·8-s + 9-s + 0.338·10-s + 1.61·12-s + 1.82·13-s − 0.338·14-s + 0.209·15-s − 1.82·17-s + 1.61·18-s − 1.61·19-s + 0.338·20-s − 0.209·21-s − 1.33·23-s + 0.999·24-s − 0.956·25-s + 2.95·26-s + 27-s − 0.338·28-s + 0.338·30-s + 1.33·31-s − 1.00·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3639\)    =    \(3 \cdot 1213\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 3639,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 4.383593684\] \[L(1,\rho) \approx 3.163166912\]

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