  In a strongly stratified turbulent layer, a uniform horizontal magnetic field
can become unstable to spontaneously form local flux concentrations due to a
negative contribution of turbulence to the large-scale (mean-field) magnetic
pressure. This mechanism, called the negative effective magnetic pressure
instability (NEMPI), is of interest in connection with dynamo scenarios where
most of the magnetic field resides in the bulk of the convection zone, and not
at the bottom. Recent work using the mean-field hydromagnetic equations has
shown that NEMPI becomes suppressed at rather low rotation rates with Coriolis
numbers as low as 0.1.}{Here we extend these earlier investigations by studying
the effects of rotation both on the development of NEMPI and on the effective
magnetic pressure. We also quantify the kinetic helicity from direct numerical
simulations (DNS) and compare with earlier work.}{To calculate the rotational
effect on the effective magnetic pressure we consider both DNS and analytical
studies using the $\tau$ approach. To study the effects of rotation on the
development of NEMPI we use both DNS and mean-field calculations of the 3D
hydromagnetic equations in a Cartesian domain.}{We find that the growth rates
of NEMPI from earlier mean-field calculations are well reproduced with DNS,
provided the Coriolis number is below about 0.06. In that case, kinetic and
magnetic helicities are found to be weak. For faster rotation, dynamo action
becomes possible. However, there is an intermediate range of rotation rates
where dynamo action on its own is not yet possible, but the rotational
suppression of NEMPI is being alleviated.}{Production of magnetic flux
concentrations through the suppression of turbulent pressure appears to be
possible only in the upper-most layers of the Sun, where the convective
turnover time is less than 2 hours.}
