# Properties

 Degree 4 Conductor $37^{3}$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 0.962·3-s − 2·4-s + 0.701·7-s − 0.185·9-s − 1.06·11-s + 1.92·12-s + 0.853·13-s + 3·16-s + 0.171·17-s − 0.821·19-s − 0.675·21-s − 0.543·23-s − 2·25-s + 0.285·27-s − 1.40·28-s + 0.845·29-s + 1.06·31-s + 1.02·33-s + 0.370·36-s + 0.00444·37-s − 0.821·39-s + 0.0342·41-s − 0.624·43-s + 2.13·44-s − 0.791·47-s − 2.88·48-s − 1.09·49-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 50653 ^{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

 $$d$$ = $$4$$ $$N$$ = $$50653$$    =    $$37^{3}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(4,\ 50653,\ (\ :1.5, 0.5),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-37^{- s})^{-1}\prod_{p \nmid 37 }\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.