Bank-Laine functions with periodic zero-sequences

A Bank-Laine function is an entire function E such that E(z) = 0 implies that E’(z) = ±1. Such functions arise as the product of linearly independent solutions of a second order linear differential equation ω″ + A(z)ω = 0 with A entire. Suppose that $$E(z)=R(z)e^{g(z)}prod_{j=1}^m prod_{k=1}^{q_j}(e^{alpha_jz}-beta_{j,k}),$$ where R is a rational function, g is a polynomial, and the α_j and β_j,k are non-zero complex numbers, and that E’(z) = ±1 at all but finally many zeros z of E. Then the quotients α_j/α_j′ are all rational numbers and E is a Bank-Laine function and reduces to the form E(z) = P₀ (e^αz) e^Q ₀(z) with α a non-zero complex number and P₀ and Q₀ polynomials.