# Properties

 Degree 5 Conductor $3^{8} \cdot 7^{2} \cdot 23^{2}$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes

# Related objects

(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

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