# Properties

 Degree $3$ Conductor $144$ 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 + 0.333·3-s − 0.200·5-s − 7-s + 0.111·9-s + 0.454·11-s − 0.692·13-s − 0.0666·15-s − 0.764·17-s − 0.157·19-s − 0.333·21-s + 1.78·23-s + 0.239·25-s + 0.0370·27-s + 0.241·29-s + 1.06·31-s + 0.151·33-s + 0.200·35-s − 0.0270·37-s − 0.230·39-s − 0.121·41-s − 0.627·43-s − 0.0222·45-s − 47-s + 2·49-s − 0.254·51-s − 0.924·53-s − 0.0909·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 144 ^{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: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $1$ Arithmetic: yes Primitive: yes Self-dual: yes Selberg data: $$(3,\ 144,\ (1:1.0),\ 1)$$

## Particular Values

$L(1/2, E, \mathrm{sym}^{2}) \approx 0.8296846420$ $L(1, E, \mathrm{sym}^{2}) \approx 0.9517316237$

## Euler product

$$L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}\prod_{p \nmid 72 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}$$