# Properties

 Degree 4 Conductor $19^{3}$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 + 0.769·3-s − 2·4-s − 0.268·5-s + 0.701·7-s + 0.814·9-s − 1.06·11-s − 1.53·12-s + 0.853·13-s − 0.206·15-s + 3·16-s + 1.07·17-s + 0.0120·19-s + 0.536·20-s + 0.540·21-s + 0.232·25-s + 1.56·27-s − 1.40·28-s − 0.845·29-s + 1.06·31-s − 0.822·33-s − 0.188·35-s − 1.62·36-s − 0.622·37-s + 0.656·39-s + 1.05·41-s + 0.301·43-s + 2.13·44-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 6859 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$6859$$    =    $$19^{3}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(4,\ 6859,\ (\ :1.5, 0.5),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-19^{- s})^{-1}\prod_{p \nmid 19 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}

## Particular Values

$L(1/2, E, \mathrm{sym}^{3}) \approx 1.201026362$ $L(1, E, \mathrm{sym}^{3}) \approx 0.9919550069$