# Properties

 Degree 4 Conductor $2^{6} \cdot 47^{3} \cdot 73^{3}$ Sign $-1$ Motivic weight 3 Primitive yes Self-dual yes

# Learn more about

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 1.73·3-s + 0.268·5-s − 0.431·7-s + 9-s + 0.657·11-s + 0.106·13-s − 0.464·15-s − 0.171·17-s + 0.821·19-s + 0.748·21-s − 1.30·23-s + 0.232·25-s − 1.32·29-s − 1.06·31-s − 1.13·33-s − 0.115·35-s + 0.324·37-s − 0.184·39-s − 0.685·41-s + 1.08·43-s + 0.268·45-s − 0.00310·47-s − 0.180·49-s + 0.296·51-s + 0.933·53-s + 0.176·55-s − 1.42·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{6} \cdot 47^{3} \cdot 73^{3}\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

 $$d$$ = $$4$$ $$N$$ = $$2^{6} \cdot 47^{3} \cdot 73^{3}$$ $$\varepsilon$$ = $-1$ primitive : yes self-dual : yes Selberg data = $(4,\ 2^{6} \cdot 47^{3} \cdot 73^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+47^{ -s})^{-1}(1-73^{- s})^{-1}\prod_{p \nmid 27448 }\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): not computed L(1): not computed

## 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.