# Properties

 Degree 3 Conductor $2^{6} \cdot 5^{4}$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s,\rho)$  = 1 + 5-s + 1.61·7-s − 11-s + 1.61·13-s − 17-s − 0.618·19-s + 25-s + 27-s − 0.618·29-s + 1.61·35-s − 41-s − 0.618·43-s + 1.61·47-s + 49-s − 55-s − 0.618·59-s + 1.61·65-s − 67-s + 1.61·71-s − 1.61·77-s − 0.618·79-s − 85-s − 89-s + 2.61·91-s − 0.618·95-s + 1.61·97-s − 0.618·103-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$3$$ $$N$$ = $$40000$$    =    $$2^{6} \cdot 5^{4}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 40000,\ (0, 1, 1:\ ),\ 1)$

## Euler product

\begin{aligned} L(s,\rho) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Particular Values

$L(1/2,\rho) \approx 1.917045898$ $L(1,\rho) \approx 1.441299733$