  Milnor's invariants are some of the more fundamental oriented link
concordance invariants; they behave as higher order linking numbers and can be
computed using combinatorial group theory (due to Milnor), Massey products (due
to Turaev and Porter), and higher order intersections (due to Cochran). In this
paper, we generalize the first non-vanishing Milnor's invariants to oriented
knots inside a closed, oriented $3$-manifold $M$. We call this the Dwyer number
of a knot and show methods to compute it for null-homologous knots inside
connected sums of $S^1 \times S^2$. We further show in this case the Dwyer
number provides the weight of the first non-vanishing Massey product in the
knot complement in the ambient manifold. Additionally, we prove the Dwyer
number detects a family of knots K in $\#^{\ell}S^1 \times S^2$ bounding
smoothly embedded disks in $\natural^{\ell} D^2 \times S^2$ which are not
concordant to the unknot.
