The Zoom has mic inputs, but also other inputs, so make sure you are using the mic inputs. These are the XLR inputs.
The XLR inputs are dual XLR and jack inputs, but if you plug in with a jack cable, I think it is actually going unbalanced to high input impedance, which would give a low level, so make sure you are using an XLR cable.
It is normal for dynamic mics to have a slightly lower output than a condenser mic, so you would have to increase the gain on the zoom more. It could be very normal to increase the gain all the way.
It is FAR better to increase the gain on the preamp to get as high level as possible (before distorting) than record quiet and increase the volume in the DAW or mixer