# Properties

 Degree 4 Conductor $3^{3} \cdot 19^{3}$ Sign $-1$ Motivic weight 3 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 0.192·3-s + 0.268·5-s − 2.96·7-s + 0.0370·9-s − 0.575·11-s − 0.938·13-s − 0.0516·15-s − 16-s + 0.470·17-s − 0.0120·19-s + 0.571·21-s + 1.08·23-s + 0.232·25-s − 0.00712·27-s + 0.691·29-s + 0.903·31-s + 0.110·33-s − 0.796·35-s + 0.180·39-s + 0.301·43-s + 0.00993·45-s + 0.363·47-s + 0.192·48-s + 4.77·49-s − 0.0906·51-s − 0.155·53-s − 0.154·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 185193 ^{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

 $$d$$ = $$4$$ $$N$$ = $$185193$$    =    $$3^{3} \cdot 19^{3}$$ $$\varepsilon$$ = $-1$ primitive : yes self-dual : yes Selberg data = $(4,\ 185193,\ (\ :1.5, 0.5),\ -1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}(1+19^{ -s})^{-1}\prod_{p \nmid 57 }\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.