# Properties

 Degree 4 Conductor $2^{6} \cdot 5^{3}$ Sign $-1$ Motivic weight 3 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 + 0.0894·5-s − 0.431·7-s − 2·9-s − 0.657·11-s + 0.938·13-s − 0.856·17-s − 1.06·19-s − 1.08·23-s + 0.00800·25-s + 0.691·29-s − 0.0926·31-s − 0.0386·35-s − 1.01·37-s + 1.05·41-s + 0.624·43-s − 0.178·45-s − 0.968·47-s − 0.180·49-s − 1.08·53-s − 0.0588·55-s + 0.900·59-s + 0.495·61-s + 0.863·63-s + 0.0839·65-s − 1.02·67-s + 1.05·73-s + 0.284·77-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 8000 ^{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

 $$d$$ = $$4$$ $$N$$ = $$8000$$    =    $$2^{6} \cdot 5^{3}$$ $$\varepsilon$$ = $-1$ primitive : yes self-dual : yes Selberg data = $(4,\ 8000,\ (\ :1.5, 0.5),\ -1)$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-5^{- s})^{-1}\prod_{p \nmid 40 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}

## Particular Values

$L(1/2, E, \mathrm{sym}^{3}) \approx 0$ $L(1, E, \mathrm{sym}^{3}) \approx 0.6440527444$