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Well, a simple counter example is $$A(s)=\sum_{n=1}^\infty a_n n^{-s}= e^{\eta(s)},$$ where $$\eta(s)=\sum_{n=1}^\infty (-1)^{n-1} n^{-s}=(1-2^{1-s})\zeta(s).$$ This Dirichlet series is obviously meromorphic since it is in fact entire and it is also absolutely convergent on some half plane Re$(s)>c$. This entire function is not of finite order by the functional equation of the Riemann zeta-function and Stirling's formula.