# Properties

 Degree 3 Conductor $2^{2} \cdot 3^{2} \cdot 5^{2}$ 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 + 0.333·3-s + 0.250·4-s + 0.200·5-s + 0.166·6-s + 1.28·7-s + 0.125·8-s + 0.111·9-s + 0.100·10-s − 11-s + 0.0833·12-s − 0.692·13-s + 0.642·14-s + 0.0666·15-s + 0.0625·16-s + 1.11·17-s + 0.0555·18-s − 0.157·19-s + 0.0500·20-s + 0.428·21-s − 0.5·22-s − 23-s + 0.0416·24-s + 0.0400·25-s − 0.346·26-s + 0.0370·27-s + 0.321·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 900 ^{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

 $$d$$ = $$3$$ $$N$$ = $$900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 900,\ (1:1.0),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 30 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}

## Particular Values

$L(1/2, E, \mathrm{sym}^{2}) \approx 2.058072776$ $L(1, E, \mathrm{sym}^{2}) \approx 1.626452958$