# Properties

 Degree $4$ Conductor $1728$ Sign $1$ Motivic weight $3$ Arithmetic yes Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 0.192·3-s + 1.07·5-s + 0.0370·9-s − 0.657·11-s + 0.938·13-s − 0.206·15-s − 0.856·17-s + 1.06·19-s − 1.30·23-s + 0.912·25-s − 0.00712·27-s − 0.845·29-s + 0.0926·31-s + 0.126·33-s − 1.01·37-s − 0.180·39-s + 1.05·41-s − 0.993·43-s + 0.0397·45-s − 2·49-s + 0.164·51-s + 0.528·53-s − 0.706·55-s − 0.204·57-s − 0.900·59-s + 0.495·61-s + 1.00·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 1728 ^{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

 Degree: $$4$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $1$ Arithmetic: yes Primitive: yes Self-dual: yes Selberg data: $$(4,\ 1728,\ (\ :1.5, 0.5),\ 1)$$

## Particular Values

$L(1/2, E, \mathrm{sym}^{3}) \approx 1.096127984$ $L(1, E, \mathrm{sym}^{3}) \approx 1.086351677$

## Euler product

$$L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}\prod_{p \nmid 24 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}$$