# Properties

 Degree 3 Conductor $2^{4} \cdot 47^{2} \cdot 73^{2}$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 + 2·3-s + 0.800·5-s + 1.28·7-s + 2·9-s + 0.454·11-s + 0.923·13-s + 1.60·15-s + 1.11·17-s − 0.789·19-s + 2.57·21-s + 1.78·23-s − 0.160·25-s + 27-s + 1.79·29-s − 0.483·31-s + 0.909·33-s + 1.02·35-s − 0.972·37-s + 1.84·39-s + 1.43·41-s − 0.418·43-s + 1.60·45-s + 0.0212·47-s + 0.367·49-s + 2.23·51-s − 0.698·53-s + 0.363·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut &\left(2^{4} \cdot 47^{2} \cdot 73^{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

 $$d$$ = $$3$$ $$N$$ = $$2^{4} \cdot 47^{2} \cdot 73^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 2^{4} \cdot 47^{2} \cdot 73^{2} ,\ ( 1 : 1.0 ),\ 1 )$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-47^{- s})^{-1}(1-73^{- s})^{-1}\prod_{p \nmid 27448 }\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.