# Properties

 Degree 4 Conductor $2^{3} \cdot 7^{3}$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes

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## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 0.353·2-s + 0.769·3-s + 0.125·4-s − 0.272·6-s + 0.0539·7-s − 0.0441·8-s + 0.814·9-s + 0.0962·12-s + 0.853·13-s − 0.0190·14-s + 0.0156·16-s + 0.171·17-s − 0.288·18-s − 0.821·19-s + 0.0415·21-s − 0.0340·24-s − 2·25-s − 0.301·26-s + 1.56·27-s + 0.00674·28-s + 0.845·29-s + 1.06·31-s − 0.00552·32-s − 0.0605·34-s + 0.101·36-s − 0.622·37-s + 0.290·38-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 2744 ^{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

 $$d$$ = $$4$$ $$N$$ = $$2744$$    =    $$2^{3} \cdot 7^{3}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(4,\ 2744,\ (\ :1.5, 0.5),\ 1)$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 14 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}

## Particular Values

$L(1/2, E, \mathrm{sym}^{3}) \approx 1.258906161$ $L(1, E, \mathrm{sym}^{3}) \approx 1.143224753$