Properties

Degree $4$
Conductor $1777$
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  − 2-s − 3-s − 5-s + 6-s + 10-s − 13-s + 15-s + 17-s + 25-s + 26-s + 2·29-s − 30-s − 31-s + 32-s − 34-s − 37-s + 39-s − 41-s − 43-s − 50-s − 51-s − 53-s − 2·58-s − 59-s + 62-s − 64-s + 65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1777 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1777\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 1777,\ (0, 0, 1, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.07944583322\] \[L(1,\rho) \approx 0.2719789018\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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