I have coefficients $a_i\in\mathbb{R}$ for $i=1,...,p$ and a term $$\mathcal{Z}(z)=p^{-z}\sum_{j=1}^p a_j \zeta(z,jp^{-1}),\qquad \sum_{j=1}^p a_j = 1$$
and I was wondering whether I can recover the coefficients $a_i$ given that I only have knowledge $\mathcal{Z}$. I thought about using the Cauchy integral theorem along a closed curve $\gamma$ enclosing the pole $1$ of $\zeta(z,a)$ combined with some function $f_i(z)$, e.g., $f_i(z)=\zeta(z,ip^{-1})$ or $f_i(z)=\zeta(1-z,m_i p^{-1})$ for some suitable integer $0<m_i<p$, in order to obtain $$a_i\cdot C(p) = \oint_{\gamma} \mathcal{Z}(z)p^{z}f_i(z)\mathrm dz$$
for some constant $C(p)$ independent of $i$. Unfortunately, in both cases, I was not succesfull and I was hoping that maybe someone could help me in finding such a "orthogonality" property which removes the contribution of the remaining summands.