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  + 2·2-s + 3·4-s + 4·8-s − 9-s + 5·16-s − 2·18-s − 25-s − 2·29-s + 6·32-s − 3·36-s − 2·37-s + 2·43-s − 2·50-s − 4·58-s + 7·64-s + 71-s − 4·72-s − 4·74-s − 2·79-s + 81-s + 4·86-s − 3·100-s − 2·107-s + 2·109-s − 6·116-s + 121-s + 8·128-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

\[L(1/2,\rho) \approx 4.349332768\] \[L(1,\rho) \approx 3.175908597\]

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