On a conjecture of Polya

For entire functions $F(z)$ of finite lower Ritt order, defined by Dirichlet series with positive exponents, a criterion is found for
$$ \ln\sup\limits_{|t|<\infty}\big|F(\sigma+it)\big|=[1+o(1)\ln\min\limits_{|t|\leqslant h}\big|F(\sigma+it)\big|\quad(0<h<\infty) $$
to hold as $\sigma\to+\infty$ outside some exceptional set.