Kernel of finitely additive function on $\mathbf{N}$ and Borel sets

Question

Let $f: \mathcal{P}(\mathbf{N}) \to [0,1]$ be a finitely additive function such that

• $f(\mathbf{N})=1$;
• $f(X+h)=f(X)$ for all $X\subseteq \mathbf{N}$ and $h \in \mathbf{N}$;
• $f(kX)=\frac{1}{k}f(X)$ for all $X\subseteq \mathbf{N}$ and $k \in \mathbf{N}$, where $kX=\{kx: x \in X\}$.

(It can be shown in ZFC that a function $f$ of this type exists.) It is easy to see that $$ Z_f:=\{X\subseteq \mathbf{N}: f(X)=0\} $$ is not a maximal ideal (or equivalently $Z_f^c$ is not an ultrafilter): indeed $f(2\mathbf{N})=f(2\mathbf{N}+1)=1/2$ so there exists a partition of $\mathbf{N}$ in two sets not belonging to $Z_f$. Recall that ultrafilters are not analytic subsets of $\{0,1\}^\mathbf{N}$. Hence it would make sense to ask the following related question:

Question. Is it true that $Z_f$ is not a Borel subset of $\{0,1\}^\mathbf{N}$?