Properties

Degree $2$
Conductor $4645$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + 4-s − 5-s − 9-s − 2·11-s + 16-s − 2·19-s − 20-s + 25-s + 2·29-s − 36-s − 2·44-s + 45-s − 49-s + 2·55-s + 2·61-s + 64-s + 2·71-s − 2·76-s − 80-s + 81-s + 2·89-s + 2·95-s + 2·99-s + 100-s − 2·101-s + 2·116-s + 3·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4645 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4645\)    =    \(5 \cdot 929\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 4645,\ (1, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 1.168492473\] \[L(1,\rho) \approx 0.8688766707\]

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