Properties

Degree $2$
Conductor $6075$
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  + 4-s + 2·7-s − 13-s + 16-s − 19-s + 2·28-s + 2·31-s − 37-s − 43-s + 3·49-s − 52-s − 61-s + 64-s + 2·67-s − 73-s − 76-s + 2·79-s − 2·91-s − 97-s − 103-s − 109-s + 2·112-s + 121-s + 2·124-s − 127-s − 2·133-s − 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6075\)    =    \(3^{5} \cdot 5^{2}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 6075,\ (0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 2.208661682\] \[L(1,\rho) \approx 1.562761741\]

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