Construction of weight function to satisfy condition on given functional

Question

Consider the following function :

$$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$

Here, $\omega(z)$ is a weight we are going to consider

The following two conditions should meet for $\omega(z)$ :

1. $$\lim_{ y→∞}|F(x ± iy)|e^{−2πy }= 0$$ uniformly with respect to $x$

2. $$\int_0^\infty |F(x + iy) − F(x − iy)|e^{−2πy} dy<+\infty$$ for every $x≥1$ and tends to zero as $x\to\infty$.

Question : Explicit construction of $\omega(z)$ .

We can say this question is Focused version of the following question (as asked in the reason for closing down the question) : https://math.stackexchange.com/q/3570663/702232

Related but different: On properties on a certain functional

All types of suggestive comments and advices are welcome.