Properties

Degree $2$
Conductor $10575$
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  − 1.95·2-s + 2.82·4-s − 1.33·7-s − 3.57·8-s + 2.61·14-s + 4.16·16-s + 1.82·17-s − 3.78·28-s − 4.57·32-s − 3.57·34-s + 1.61·37-s + 47-s + 0.790·49-s + 0.618·53-s + 4.78·56-s − 1.33·59-s + 0.618·61-s + 4.78·64-s + 5.16·68-s + 0.209·71-s − 3.16·74-s − 1.61·79-s − 83-s − 0.618·89-s − 1.95·94-s − 0.618·97-s − 1.54·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10575 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10575\)    =    \(3^{2} \cdot 5^{2} \cdot 47\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 10575,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.4909029848\] \[L(1,\rho) \approx 0.4393299001\]

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