Dirichlet series of convolution of conditionally convergent Dirichlet series is not necessarily convergent

Question

In Montgomery and Vaughan's Multiplicative Number Theory I. Classical Theory, they claim the statement in this question's title, using the example $$ \alpha(s) = \sum_{n=1}^\infty (-1)^{n-1}n^{-s} $$ as the conditionally convergent series, for $0< \sigma < 1$, and state that the Dirichlet series of $\alpha(s)^2$ has abscissa of convergence $1/4$. However, I cannot see why the abscissa of convergence is $1/4$.