Properties

Degree $2$
Conductor $119025$
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 − 2·11-s + 16-s + 2·31-s + 2·44-s − 49-s − 64-s − 2·89-s + 3·121-s − 2·124-s + 2·139-s − 2·149-s − 2·151-s − 169-s − 2·176-s + 2·191-s + 196-s + 2·211-s + 2·251-s + 256-s − 2·271-s − 2·281-s − 289-s − 2·331-s − 4·341-s − 2·349-s + 2·356-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119025 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119025\)    =    \(3^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 119025,\ (1, 1:\ ),\ -1)\)

Particular Values

\[L(1/2,\rho) \approx 0\] \[L(1,\rho) \approx 0.6865811757\]

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