# Properties

 Degree 3 Conductor $11^{2}$ 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 − 0.666·3-s − 0.800·5-s − 0.666·6-s − 0.428·7-s + 1.11·9-s − 0.800·10-s + 0.0909·11-s + 0.230·13-s − 0.428·14-s + 0.533·15-s + 16-s − 0.764·17-s + 1.11·18-s − 19-s + 0.285·21-s + 0.0909·22-s − 0.956·23-s + 1.43·25-s + 0.230·26-s − 0.185·27-s − 29-s + 0.533·30-s + 0.580·31-s + 32-s − 0.0606·33-s − 0.764·34-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 121 ^{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

 $$d$$ = $$3$$ $$N$$ = $$121$$    =    $$11^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 121,\ (1:1.0),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1+2\ 3^{- s}-6 \ 3^{-2 s}-27 \ 3^{-3 s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 99 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}

## Particular Values

$L(1/2, E, \mathrm{sym}^{2}) \approx 0.8933960461$ $L(1, E, \mathrm{sym}^{2}) \approx 1.057599244$