Properties

Degree $2$
Conductor $3479$
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 − 1.94·3-s + 2.24·4-s + 1.56·5-s + 3.51·6-s − 2.24·8-s + 2.80·9-s − 2.81·10-s − 4.38·12-s − 3.04·15-s + 1.80·16-s − 5.04·18-s + 0.867·19-s + 3.51·20-s + 4.38·24-s + 1.44·25-s − 3.51·27-s + 0.445·29-s + 5.49·30-s − 0.999·32-s + 6.29·36-s + 1.80·37-s − 1.56·38-s − 3.51·40-s + 1.24·43-s + 4.38·45-s − 3.51·48-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

\[L(1/2,\rho) \approx 0.4238515687\] \[L(1,\rho) \approx 0.4031892398\]

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