$\lim_{z \to x \pm i \infty} \Gamma(z) \zeta(z + \alpha) = 0$?

Question

I guess $\lim_{z \to x \pm i \infty} \Gamma(z) \zeta(z + \alpha) = 0$ where $x$ and $\alpha$ are real numbers. The guessing is from numerical experiments and I know $\Gamma(z)$ vanishes exponentially in the limit. However I have no idea about the behavior of the Riemann zeta function in the limit. I'd like to know whether the conjecture is right and about the behavior of the Riemann zeta function in the limit.