Properties

Degree 5
Conductor $ 3^{8} \cdot 7^{2} \cdot 23^{2} $
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  − 3-s + 9-s − 19-s + 23-s − 27-s − 29-s + 31-s + 32-s − 37-s − 43-s + 57-s − 59-s + 61-s − 69-s + 71-s − 73-s + 81-s + 87-s + 89-s − 93-s − 96-s − 97-s − 103-s − 107-s − 109-s + 111-s + 129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{2} \cdot 23^{2}\right)^{s/2} \, \Gamma_{\R}(s)^{5} \, L(s,\rho)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(3^{8} \cdot 7^{2} \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 3^{8} \cdot 7^{2} \cdot 23^{2} ,\ ( 0, 0, 0, 0, 0 : \ ),\ 1 )$

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{5} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.2803543705\] \[L(1,\rho) \approx 0.6810544886\]

Imaginary part of the first few zeros on the critical line

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