Study number of complex zeros of a function contained in a strip

Question

Consider the equation $$8\pi e^{-2}z-e^{-2z}+4E_{1}(2z)=0$$ ($E_{1}$ is the exponential integral of order 1).

I would like to study the number of solutions in the strip $0<\Re(z)\le 1$ (and possibly also determine zones of the strip in which only a solution exists). I was able to prove that all the solution must be in the intersection of the vertical strip with a horizontal strip $-\alpha\le\Im(z)\le\alpha$ for some $\alpha>0$ finite. To prove this i considered $w=a+ib$ with $0<a\le\beta$ fixed. If i let $|b|\to +\infty$ i have (since $|\Re(z)|$ is bounded) that

1. The first term's absolute value diverges to $+\infty$
2. The second term has constant absolute value equal to $e^{-2a}$
3. The third term goes to zero

This means that for $|b|$ large enough there cannot be solutions. This is valid independently on the $0<a\le\beta$ fixed that has been chosen.

Now, how can I get to know if the solutions in this rectangle are a finite quantity or not. And is it possible to have a bit more information on their location? Moreover, does the way to tackle the problem change if I add a constant term in the equation?

Thanks a lot in advance.

EDIT: Can I use the identity principle to conclude that if the zeros are an infinite quantity, then they can only accumulate near 0 or otherwise the function would be identically zero?