Confusion about exponentiating differential operators

Question

I am studying the Conformal group on the Riemann sphere and trying to verify that the generators do indeed produce the transformation they are claimed to produce. I am running into confusion as to what calculation I am supposed to do. Consider these two differential operators $ L_1 = c \frac{d }{dz}$ and $L_2 =cz \frac{d}{dz}$, where c is some real constant. These are two of generators of the lie algebra and formally speaking I am supposed to exponentiate them in order to see how they act on $z$. So we have $e^{c\frac{d}{dz}}(z)$ and $e^{cz\frac{d}{dz}}(z)$. The question is how I interpret the exponential. For $L_1$ I took it literally and calculated $\left(\sum_{n=0}^{n=\infty} \frac{c^n}{n!} \frac{d^n}{dz^n} \right)z$ then I got the translation I expected namely $z \rightarrow z + c $. This failed when I tried $e^{cz\frac{d}{dz}}(z)$ because obviously the expression got has only two terms and one does not get $z \rightarrow cz $ so instead I treated $e^{cz\frac{d}{dz}}(z)$ as a symbolic expression to be calculated as meaning $ \sum_{n=0}^{n=\infty}\frac{ad_{cz\frac{d}{dz}}^n}{n!}(z)$ where $ad_{cz\frac{d}{dz}}(z) = [c\frac{d}{dz}, z]$ which gives $z \rightarrow e^c z$ which is certainly a dilation I expect. This seems to be a case where it is necessary to realize the differential operators are really a basis for a lie algebra but I can't seem to put all the pieces completely together neatly.