Properties

Degree $3$
Conductor $3775$
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  + 1.61·2-s + 4-s − 5-s − 1.61·7-s − 1.61·10-s − 11-s + 0.618·13-s − 2.61·14-s − 20-s − 1.61·22-s + 25-s + 0.999·26-s − 27-s − 1.61·28-s − 29-s + 1.61·31-s + 32-s + 1.61·35-s − 0.618·37-s + 0.618·41-s − 44-s − 0.618·47-s + 49-s + 1.61·50-s + 0.618·52-s + 0.618·53-s − 1.61·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(3775\)    =    \(5^{2} \cdot 151\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 3775,\ (0, 0, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.7020278521\] \[L(1,\rho) \approx 1.269275152\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line