# Properties

 Degree 3 Conductor 1369 Sign $1$ Self-dual yes Motivic weight 2

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 − 2-s − 0.666·3-s + 2·4-s − 5-s + 0.666·6-s − 0.857·7-s − 2·8-s + 1.111·9-s + 10-s − 0.181·11-s − 1.333·12-s + 0.230·13-s + 0.857·14-s + 0.666·15-s + 3·16-s + 1.117·17-s − 1.111·18-s − 0.789·19-s − 2·20-s + 0.571·21-s + 0.181·22-s + 0.565·23-s + 1.333·24-s + 2·25-s − 0.230·26-s − 0.185·27-s − 1.714·28-s + ⋯

## Functional equation

\begin{align} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1369 ^{s/2} \Gamma_{\R}(s+1) \Gamma_{\C}(s+1) \cdot L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{align}

## Invariants

 $d$ = $3$ $N$ = $1369$    =    $37^{2}$ $\varepsilon$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 1369,\ (1:1.0),\ 1)$

## Euler product

$$$L(s, E, \mathrm{sym}^{2}) = (1-37^{- s})^{-1}\prod_{p \nmid 37 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}$$$

## Particular Values

$L(1/2, E, \mathrm{sym}^{2}) \approx 0.7728058998$ $L(1, E, \mathrm{sym}^{2}) \approx 0.6534792516$