Let $G=SO_n$ and fix a borel subgroup $P_0$ of $G$. Let $P=MN$ be a standard maximal parabolic subgroup $G$ and $\sigma$ a cuspidal representation of $M$

Consider the normalized parabolic induced representation $\text{Ind}_P^G(\sigma|\cdot|^z)$ and for sufficiently large $z$, we can define Eisenstein series $E(z,\phi)$ for $\phi \in \text{Ind}_P^G(\sigma)$. Since $E(z,\phi)$ has a meromorphic continuation, let $z_0$ be a simple pole of $E(z,\phi)$. Put $\mathcal{E}(\phi,z_0)$ a residue of $E(z,\phi)$ at $z=z_0$.

I am wondering if there are two $\phi_1,\phi_2 \in \text{Ind}_P^G(\sigma)$ such that $\mathcal{E}(\phi_1,z_0)=\mathcal{E}(\phi_2,z_0)$, then $E(\phi_1,z)=E(\phi_2,z)$ as meromorphic functions on $\mathbb{C}$. Is it right?