# Properties

 Degree 3 Conductor $3^{2} \cdot 19^{2}$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 + 2-s + 0.333·3-s + 0.800·5-s + 0.333·6-s + 2.57·7-s + 0.111·9-s + 0.800·10-s − 0.909·11-s − 0.692·13-s + 2.57·14-s + 0.266·15-s + 16-s − 0.941·17-s + 0.111·18-s + 0.0526·19-s + 0.857·21-s − 0.909·22-s − 0.304·23-s − 0.160·25-s − 0.692·26-s + 0.0370·27-s − 0.862·29-s + 0.266·30-s + 0.161·31-s + 32-s − 0.303·33-s − 0.941·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 3249 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}

## Invariants

 $$d$$ = $$3$$ $$N$$ = $$3249$$    =    $$3^{2} \cdot 19^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $$(3,\ 3249,\ (1:1.0),\ 1)$$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-19^{- s})^{-1}\prod_{p \nmid 57 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}

## Particular Values

$L(1/2, E, \mathrm{sym}^{2}) \approx 3.841181158$ $L(1, E, \mathrm{sym}^{2}) \approx 2.349533923$