Let $D(s)$ be a Dirichlet series with abscissa of convergence $\sigma_c=\sigma_a$. Does it follow that the Dirichlet series defined by $P(s)=D(s)\bar D(\bar s)$ has the same abscissa of convergence?

In light of Noam D. Elkies answer below, I would like to know about conditions on the coefficients that also lead to the divergence of the square? What if the coefficients are completely multiplicative?