Properties

Degree $3$
Conductor $18571$
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  − 7-s − 8-s + 13-s − 27-s + 29-s − 41-s + 43-s + 49-s + 56-s + 64-s + 71-s + 3·83-s − 91-s − 97-s − 104-s + 113-s + 125-s − 127-s + 3·139-s + 3·167-s + 2·169-s + 181-s + 189-s − 197-s − 203-s − 211-s + 216-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18571 ^{s/2} \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(18571\)    =    \(7^{2} \cdot 379\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 18571,\ (1, 1, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 1.281623430\] \[L(1,\rho) \approx 0.9101059725\]

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