Does every pole have non-zero residue?

Question

I am self-studying complex analysis.I am currently studying isolated singularity.There are $3$ kinds of singularities viz. removable singularity,pole and essential singularity.We know that if $z_0$ is a pole of $f$ then $f(z)=\frac{a_{-n}}{(z-z_0)^n}+\frac{a_{-(n-1)}}{(z-z_0)^{n-1}}+\dots+\frac{a_{-1}}{z-z_0}+H(z)$ where $H$ is a non vanishing holomorphic function in a neighborhood of $z_0$.$a_{-1}$ is called the residue of $f$ at $z_0$.My question is that can it happen that residue is $0$ but yet the function has a pole at $z_0$.I think the answer is yes,because we can take for example $f(z)=\frac{2}{z^2}+\sin(z)$ where $z\in \mathbb C-\{0\}$.Please tell me if I am correct.Also tell me what is the meaning of a zero residue?