# Properties

 Degree $3$ Conductor $225$ Sign $1$ Motivic weight $2$ Arithmetic yes Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 − 0.5·2-s + 0.333·3-s + 0.750·4-s + 0.200·5-s − 0.166·6-s − 7-s + 0.375·8-s + 0.111·9-s − 0.100·10-s + 0.454·11-s + 0.250·12-s − 0.692·13-s + 0.5·14-s + 0.0666·15-s − 0.312·16-s − 0.764·17-s − 0.0555·18-s − 0.157·19-s + 0.149·20-s − 0.333·21-s − 0.227·22-s − 23-s + 0.125·24-s + 0.0400·25-s + 0.346·26-s + 0.0370·27-s − 0.750·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 225 ^{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

 Degree: $$3$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $1$ Arithmetic: yes Primitive: yes Self-dual: yes Selberg data: $$(3,\ 225,\ (1:1.0),\ 1)$$

## Particular Values

$L(1/2, E, \mathrm{sym}^{2}) \approx 0.8889069617$ $L(1, E, \mathrm{sym}^{2}) \approx 0.9364885434$

## Euler product

$$L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 225 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}$$