# Properties

 Degree $4$ Conductor $54872$ 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 − 0.353·2-s − 0.962·3-s + 0.125·4-s + 0.340·6-s + 0.701·7-s − 0.0441·8-s − 0.185·9-s − 2.30·11-s − 0.120·12-s − 0.106·13-s − 0.248·14-s + 0.0156·16-s − 1.07·17-s + 0.0654·18-s + 0.0120·19-s − 0.675·21-s + 0.814·22-s − 1.00·23-s + 0.0425·24-s − 2·25-s + 0.0377·26-s + 0.285·27-s + 0.0877·28-s + 1.32·29-s + 1.06·31-s − 0.00552·32-s + 2.21·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 54872 ^{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: $$54872$$    =    $$2^{3} \cdot 19^{3}$$ Sign: $1$ Arithmetic: yes Primitive: yes Self-dual: yes Selberg data: $$(4,\ 54872,\ (\ :1.5, 0.5),\ 1)$$

## Particular Values

L(1/2): not computed L(1): not computed

## Euler product

$$L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1-19^{- s})^{-1}\prod_{p \nmid 38 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}$$

## Imaginary part of the first few zeros on the critical line

Zeros not available.

## Graph of the $Z$-function along the critical line

Plot not available.