Properties

Degree $2$
Conductor $1205$
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 + 16-s + 20-s + 25-s − 2·29-s + 36-s + 2·41-s + 45-s − 49-s − 2·59-s − 2·61-s + 64-s + 2·79-s + 80-s + 81-s + 100-s − 2·116-s + 121-s + 125-s + 144-s − 2·145-s + 2·151-s + 2·164-s − 169-s + 180-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 1205,\ (1, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 2.887952100\] \[L(1,\rho) \approx 1.705915331\]

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