# Properties

 Degree 4 Conductor $3^{4} \cdot 7^{3}$ Sign $-1$ Motivic weight 3 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 1.06·2-s + 0.375·4-s − 1.07·5-s − 0.0539·7-s − 0.662·8-s + 1.13·10-s + 0.657·11-s + 0.938·13-s + 0.0572·14-s + 0.546·16-s + 0.171·17-s − 1.06·19-s − 0.402·20-s − 0.697·22-s + 0.912·25-s − 0.995·26-s − 0.0202·28-s − 0.691·29-s + 0.580·32-s − 0.181·34-s + 0.0579·35-s − 1.01·37-s + 1.12·38-s + 0.711·40-s + 0.594·41-s + 0.993·43-s + 0.246·44-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 27783 ^{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

 $$d$$ = $$4$$ $$N$$ = $$27783$$    =    $$3^{4} \cdot 7^{3}$$ $$\varepsilon$$ = $-1$ primitive : yes self-dual : yes Selberg data = $(4,\ 27783,\ (\ :1.5, 0.5),\ -1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+7^{ -s})^{-1}\prod_{p \nmid 63 }\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.