# Properties

 Degree 3 Conductor $2^{2} \cdot 13873^{2}$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 + 0.5·2-s + 0.333·3-s + 0.250·4-s − 0.200·5-s + 0.166·6-s + 1.28·7-s + 0.125·8-s − 0.222·9-s − 0.100·10-s + 1.27·11-s + 0.0833·12-s + 0.230·13-s + 0.642·14-s − 0.0666·15-s + 0.0625·16-s + 0.470·17-s − 0.111·18-s + 1.57·19-s − 0.0500·20-s + 0.428·21-s + 0.636·22-s + 0.565·23-s + 0.0416·24-s + 0.239·25-s + 0.115·26-s + 0.814·27-s + 0.321·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut &\left(2^{2} \cdot 13873^{2}\right)^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}

## Invariants

 $$d$$ = $$3$$ $$N$$ = $$2^{2} \cdot 13873^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 2^{2} \cdot 13873^{2} ,\ ( 1 : 1.0 ),\ 1 )$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-13873^{- s})^{-1}\prod_{p \nmid 27746 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-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.