Let $M$ be a renormalization/momentum scale, $\lambda$ a coupling, $G^{(n)}$ the $n$-point Green's function, $Z$ the field strength, and $\Lambda$ a momentum cutoff.
When studying the renormalization group we define the following: $$\beta \equiv M\frac{\partial}{\partial M}\lambda\big|_{\lambda_0, \Lambda}$$ and $$\gamma(\lambda) \equiv \frac{1}{2}\frac{M}{Z}\frac{\partial}{\partial M} Z.$$
On page 417 of Peskin & Schroeder, they give a nice interpretation of the $\beta$ function:
...the $\beta$ function is the rate of change of the renormalized coupling at the scale $M$ corresponding to a fixed bare coupling.
For this interpretation to hold wouldn't it make sense to define $\beta$ without the $M$ in front? i.e. $$\beta \equiv \frac{\partial}{\partial M}\lambda\big|_{\lambda_0, \Lambda}.$$
I am much more lost on the interpretation of the $\gamma$ function. I suspect it is something along the lines of measuring how the field strength changes with the momentum scale, but if this is true why does $\gamma$ take the coupling $\lambda$ as an input? Furthermore, why is there a factor $\frac12 \frac{M}{Z}$ in front?
