Properties

Degree 4
Conductor $ 19 \cdot 151 $
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  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 17-s − 20-s + 21-s + 22-s + 26-s − 28-s − 30-s + 33-s − 34-s + 35-s − 37-s + 39-s + 41-s − 42-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2869 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2869\)    =    \(19 \cdot 151\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((4,\ 2869,\ (0, 0, 1, 1:\ ),\ 1)\)

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.1029906673\] \[L(1,\rho) \approx 0.3186572303\]

Imaginary part of the first few zeros on the critical line

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