# Properties

 Degree 4 Conductor $2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3}$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 + 0.353·2-s + 0.192·3-s + 0.125·4-s − 0.0894·5-s + 0.0680·6-s + 0.0539·7-s + 0.0441·8-s + 0.0370·9-s − 0.0316·10-s + 0.0240·12-s − 0.938·13-s + 0.0190·14-s − 0.0172·15-s + 0.0156·16-s − 0.171·17-s + 0.0130·18-s + 1.06·19-s − 0.0111·20-s + 0.0103·21-s + 0.00850·24-s + 0.00800·25-s − 0.331·26-s + 0.00712·27-s + 0.00674·28-s + 0.845·29-s − 0.00608·30-s + 1.06·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 9261000 ^{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

 $$d$$ = $$4$$ $$N$$ = $$9261000$$    =    $$2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(4,\ 9261000,\ (\ :1.5, 0.5),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1+5^{ -s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 210 }\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.