# Properties

 Degree $4$ Conductor $9261$ 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 + 1.06·2-s + 0.192·3-s + 0.375·4-s + 1.07·5-s + 0.204·6-s − 0.0539·7-s + 0.662·8-s + 0.0370·9-s + 1.13·10-s − 0.657·11-s + 0.0721·12-s + 0.938·13-s − 0.0572·14-s + 0.206·15-s + 0.546·16-s − 0.171·17-s + 0.0392·18-s − 1.06·19-s + 0.402·20-s − 0.0103·21-s − 0.697·22-s + 0.127·24-s + 0.912·25-s + 0.995·26-s + 0.00712·27-s − 0.0202·28-s + 0.691·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 9261 ^{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: $$9261$$    =    $$3^{3} \cdot 7^{3}$$ Sign: $1$ Arithmetic: yes Primitive: yes Self-dual: yes Selberg data: $$(4,\ 9261,\ (\ :1.5, 0.5),\ 1)$$

## Particular Values

$L(1/2, E, \mathrm{sym}^{3}) \approx 3.054031827$ $L(1, E, \mathrm{sym}^{3}) \approx 2.137044968$

## Euler product

$$L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}(1+7^{ -s})^{-1}\prod_{p \nmid 21 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}$$