# Properties

 Degree $4$ Conductor $3.867\times 10^{12}$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 + 0.353·2-s + 0.192·3-s + 0.125·4-s + 0.0680·6-s + 0.0539·7-s + 0.0441·8-s + 0.0370·9-s + 0.0240·12-s + 0.938·13-s + 0.0190·14-s + 0.0156·16-s + 0.0130·18-s + 2.51·19-s + 0.0103·21-s + 0.00850·24-s + 0.331·26-s + 0.00712·27-s + 0.00674·28-s + 0.845·29-s − 1.06·31-s + 0.00552·32-s + 0.00462·36-s − 1.15·37-s + 0.887·38-s + 0.180·39-s − 1.05·41-s + 0.00367·42-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{4} \cdot 7^{3} \cdot 17^{4}\right)^{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: $$2^{3} \cdot 3^{3} \cdot 5^{4} \cdot 7^{3} \cdot 17^{4}$$ Sign: $1$ Primitive: yes Self-dual: yes Selberg data: $$(4,\ 2^{3} \cdot 3^{3} \cdot 5^{4} \cdot 7^{3} \cdot 17^{4} ,\ ( \ : 1.5, 0.5 ),\ 1 )$$

## Particular Values

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

## Euler product

$$L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1} (1-7^{- s})^{-1} \prod_{p \nmid 303450 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}$$

## 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.