# Properties

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

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## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 + 0.5·2-s − 3-s + 0.250·4-s + 0.800·5-s − 0.5·6-s + 1.28·7-s + 0.125·8-s + 9-s + 0.400·10-s − 11-s − 0.250·12-s − 0.923·13-s + 0.642·14-s − 0.800·15-s + 0.0625·16-s − 0.470·17-s + 0.5·18-s − 0.157·19-s + 0.200·20-s − 1.28·21-s − 0.5·22-s − 23-s − 0.125·24-s − 0.160·25-s − 0.461·26-s − 27-s + 0.321·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 324 ^{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

 $$d$$ = $$3$$ $$N$$ = $$324$$    =    $$2^{2} \cdot 3^{4}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 324,\ (1:1.0),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1+3\ 3^{- s})^{-1}\prod_{p \nmid 162 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}

## Particular Values

$L(1/2, E, \mathrm{sym}^{2}) \approx 1.181496032$ $L(1, E, \mathrm{sym}^{2}) \approx 1.153866273$