  We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve
$C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$.
We define a degree 9 polynomial $\psi_f\in \mathbb{Q}[x]$ such that the
splitting field of $\psi_f(x^3/2)$ is the $2$-torsion field of the Jacobian of
$C$. We prove that, for all but a density zero subset of primes, the zeta
function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of
$C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $\psi_f$; we
also show that for primes $\equiv 1 \pmod{3}$ the matrix $A_p$ suffices and
that for primes $\equiv 2 \pmod{3}$ the genericity assumption on $C$ is
unnecessary. An element of the proof, which may be of independent interest, is
the determination of the density of the set of primes of ordinary reduction for
a generic Picard curve. By combining this with recent work of Sutherland, we
obtain a practical deterministic algorithm that computes $Z(C_p,T)$ for almost
all primes $p \le N$ using $N\log(N)^{3+o(1)}$ bit operations. This is the
first practical result of this type for curves of genus greater than 2.
