  Scenarios for multi-component scalar dark matter based on a single $Z_N$
($N\geq 4$) symmetry are simple and well-motivated. In this paper we
investigate, for the first time, the phenomenology of the $Z_5$ model for
two-component dark matter. This model, which can be seen as an extension of the
well-known singlet scalar model, features two complex scalar fields--the dark
matter particles--that are Standard Model singlets but have different charges
under a $Z_5$ symmetry. The interactions allowed by the $Z_5$ give rise to
novel processes between the dark matter particles that affect their relic
densities and their detection prospects, which we study in detail. The key
parameters of the model are identified and its viable regions are characterized
by means of random scans. We show that, unlike the singlet scalar model, dark
matter masses below the TeV are still compatible with present data. Even though
the dark matter density turns out to be dominated by the lighter component, we
find that current and future direct detection experiments may be sensitive to
signals from both dark matter particles.
