# Properties

 Degree 4 Conductor 7112448 Sign $1$ Self-dual yes Motivic weight 3

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 1.073·5-s + 0.053·7-s − 0.657·11-s + 0.938·13-s + 0.171·17-s + 1.062·19-s + 0.912·25-s − 0.691·29-s − 0.057·35-s − 1.013·37-s + 0.594·41-s − 0.993·43-s + 0.002·49-s + 1.088·53-s + 0.706·55-s + 0.688·59-s + 0.495·61-s − 1.007·65-s + 0.860·67-s + 1.058·73-s − 0.035·77-s + 2.233·79-s + 0.349·83-s − 0.183·85-s + 0.300·89-s + 0.050·91-s − 1.140·95-s + ⋯

## Functional equation

\begin{align} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 7112448 ^{s/2} \Gamma_{\C}(s+1.5) \Gamma_{\C}(s+0.5) \cdot L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{align}

## Invariants

 $d$ = $4$ $N$ = $7112448$    =    $2^{8} \cdot 3^{4} \cdot 7^{3}$ $\varepsilon$ = $1$ primitive : yes self-dual : yes Selberg data = $(4,\ 7112448,\ (\ :1.5, 0.5),\ 1)$

## Euler product

$$$L(s, E, \mathrm{sym}^{3}) = (1-7^{- s})^{-1}\prod_{p \nmid 1008 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}$$$

## 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.