shimura varieties - What is the image of complex conjugation under Siegel Galois representations?

Let $G$ be the reductive group $operatorname{GSp}_{4}$. Let $pi$ be a smooth admissible cuspidal representation of $operatorname{GSp}_{4}(mathbb{A}^{(infty)})$ of dominant weight. Assume, for caution, that $pi$ satisfies a multiplicity one hypothesis.

Fix $p$ an odd prime. To $pi$ is attached a $p$-adic representation $rho$ of the absolute Galois group of $mathbb{Q}$ unramified outside a finite set of finite places and such that the characteristic polynomial of the Frobenius morphisms $Frell$ for $ell$ outside this set coincides with the Euler factor at $ell$ of the degree 4 $L$-function of $pi$. This Galois representation occurs in the degree 3 cohomology of the étale cohomology of a Siegel-Shimura variety.

The image of complex conjugation under $rho$ is semi-simple so can be chosen to be diagonal with eigenvalues 1 and -1. How many $-1$ are there?