Hi,
The radiated wave in far field has an eletrical field E that:
a) is perpendicular to the direction of propagation
b) parallel to E must be a component of charge acceleration; this needs a component of current parallel to E
You are asking for the direction theta=0, i.e. along the z axis.
By a), E must be in the xy plane But as the current in the dipole is along z, not having any component in x nor y, by b) Ex=Ey=0.
So, there is no electrical radiated field along theta=0.
Z