# Properties

 Degree 3 Conductor $2^{2} \cdot 3^{4} \cdot 11^{2}$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s,\rho)$  = 1 − 2-s + 4-s + 1.61·5-s − 8-s − 1.61·10-s + 11-s + 1.61·13-s + 16-s − 0.618·17-s + 1.61·20-s − 22-s − 0.618·23-s + 25-s − 1.61·26-s − 32-s + 0.618·34-s − 37-s − 1.61·40-s − 43-s + 44-s + 0.618·46-s + 1.61·47-s − 50-s + 1.61·52-s + 1.61·55-s + 1.61·61-s + 64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 39204 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$3$$ $$N$$ = $$39204$$    =    $$2^{2} \cdot 3^{4} \cdot 11^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $$(3,\ 39204,\ (0, 1, 1:\ ),\ 1)$$

## Euler product

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

## Particular Values

$L(1/2,\rho) \approx 1.310066331$ $L(1,\rho) \approx 1.018078649$