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  + 1.24·2-s + 0.867·3-s + 0.554·4-s + 1.94·5-s + 1.08·6-s − 0.554·8-s − 0.246·9-s + 2.43·10-s + 0.481·12-s + 1.69·15-s − 1.24·16-s − 0.307·18-s − 1.56·19-s + 1.08·20-s − 0.481·24-s + 2.80·25-s − 1.08·27-s + 1.80·29-s + 2.10·30-s − 1.00·32-s − 0.137·36-s − 1.24·37-s − 1.94·38-s − 1.08·40-s − 0.445·43-s − 0.481·45-s − 1.08·48-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

\[L(1/2,\rho) \approx 3.746146552\] \[L(1,\rho) \approx 2.751011544\]

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