Properties

Degree $2$
Conductor $3479$
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 − 1.24·3-s − 0.801·4-s + 1.80·5-s + 0.554·6-s + 0.801·8-s + 0.554·9-s − 0.801·10-s + 1.00·12-s − 2.24·15-s + 0.445·16-s − 0.246·18-s + 0.445·19-s − 1.44·20-s − 1.00·24-s + 2.24·25-s + 0.554·27-s + 1.24·29-s + 1.00·30-s − 1.00·32-s − 0.445·36-s − 0.445·37-s − 0.198·38-s + 1.44·40-s − 1.80·43-s + 1.00·45-s − 0.554·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3479\)    =    \(7^{2} \cdot 71\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 3479,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.7583483093\] \[L(1,\rho) \approx 0.6870137853\]

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