# Properties

 Degree 3 Conductor $2^{6} \cdot 431^{2}$ 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 + 1.27·11-s + 0.230·13-s + 1.60·15-s + 1.11·17-s + 1.57·19-s + 2.57·21-s − 0.608·23-s − 0.160·25-s + 27-s − 0.689·29-s − 0.483·31-s + 2.54·33-s + 1.02·35-s + 0.729·37-s + 0.461·39-s − 0.121·41-s + 0.488·43-s + 1.60·45-s − 0.234·47-s + 0.367·49-s + 2.23·51-s − 0.981·53-s + 1.01·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 11888704 ^{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

 $$d$$ = $$3$$ $$N$$ = $$11888704$$    =    $$2^{6} \cdot 431^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 11888704,\ (1:1.0),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-431^{- s})^{-1}\prod_{p \nmid 27584 }\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.