A field $\phi(z)$ has the conformal weight $h$, if it transforms under $z\rightarrow z_1(z)$ as
$$ \phi(z) = \tilde{\phi}(z_1)\left(\frac{dz_1}{dz}\right)^h $$
The (classical) scaling dimension can be obtained for each field by appearing in the Lagrangian by making use of the constraint that has to be dimensionless, resulting for example in
$$ [\phi] = [A^{\mu}] = 1 $$
for a scalar and a gauge field or
$$ [\Psi_D] = [\Psi_M] = [\chi] = [\eta] = \frac{3}{2} $$
for Dirac, Majorana, and Weyl spinors.
Are these two concepts of scaling dimension and conformal weight somehow related?