Properties

Degree 4
Conductor $ 5 \cdot 643 $
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  − 5-s − 7-s − 16-s + 23-s + 29-s − 31-s + 35-s + 2·47-s + 49-s − 53-s + 80-s − 81-s − 83-s − 89-s + 97-s − 101-s + 2·107-s + 112-s − 115-s − 2·127-s + 131-s − 139-s − 145-s − 149-s + 155-s − 161-s + 193-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3215\)    =    \(5 \cdot 643\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3215,\ (0, 1, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.3786092567\] \[L(1,\rho) \approx 0.6887413040\]

Imaginary part of the first few zeros on the critical line

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