  The magnetic properties of the Zn-substituted overdoped high-$T_c$
superconductor La$_{2-x}$Sr$_{x}$Cu$_{1-y}$Zn$_{y}$O$_{4}$ have been studied by
magnetization measurements and neutron scattering. Magnetization measurements
reveal that for Zn-free samples with $x \geq 0.22$ a Curie term is induced in
the temperature dependence of the magnetic susceptibility implying the
existence of local paramagnetic moments. The induced Curie constant corresponds
to a moment of 0.5 $\mu_B$ per additional Sr$^{2+}$ ion that exceeds $x=0.22$.
Zn-substitution in the overdoped \lsco also induces a Curie term that
corresponds to 1.2 $\mu_B$ per Zn$^{2+}$ ion, simultaneously suppressing $T_c$.
The relationship between $T_c$ and the magnitude of the Curie term for Zn-free
\lsco with $x \geq 0.22$ and for Zn-substituted \lsco with $x = 0.22$ are
closely similar. This signifies a general competitive relationship between the
superconductivity and the induced paramagnetic moment. Neutron scattering
measurements show that Zn-substitution in overdoped \lsco anomalously enhances
the inelastic magnetic scattering spectra around the $(\pi, \pi)$ position,
peaking at $\omega \sim 7$ meV. These facts are discussed on the basis of a
"swiss-cheese" model of Zn-substituted systems as well as a microscopic phase
separation scenario in the overdoped region indicated by muon-spin-relaxation
measurements.
