# Properties

 Degree 4 Conductor 3375 Sign $1$ Self-dual yes Motivic weight 3

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 + 1.060·2-s − 0.192·3-s + 0.375·4-s + 0.089·5-s − 0.204·6-s + 0.662·8-s + 0.037·9-s + 0.094·10-s + 0.657·11-s − 0.072·12-s + 0.938·13-s − 0.017·15-s + 0.546·16-s − 0.856·17-s + 0.039·18-s − 1.062·19-s + 0.033·20-s + 0.697·22-s − 0.127·24-s + 0.008·25-s + 0.995·26-s − 0.007·27-s + 0.691·29-s − 0.018·30-s − 0.580·32-s − 0.126·33-s − 0.907·34-s + ⋯

## Functional equation

\begin{align} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 3375 ^{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

 $d$ = $4$ $N$ = $3375$    =    $3^{3} \cdot 5^{3}$ $\varepsilon$ = $1$ primitive : yes self-dual : yes Selberg data = $(4,\ 3375,\ (\ :1.5, 0.5),\ 1)$

## Euler product

$$$L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 15 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}$$$

## Particular Values

$L(1/2, E, \mathrm{sym}^{3}) \approx 1.9341506528$ $L(1, E, \mathrm{sym}^{3}) \approx 1.6562561819$