It is well-known if $\sigma_a$ is abscissa of absolute convergence of $f(s)=\displaystyle \sum_{n=1}^{\infty} \dfrac{a_n}{n^{s}}$, then \begin{align*} \sigma_c \leq \sigma_a \leq \sigma_c+1 \end{align*} with $\sigma_c$ is abscissa of convergence of $f$.
I wonder given $\alpha \in (0,1)$, how can we construct a Dirichlet series with $\sigma_a=\sigma_c+\alpha$? For $\sigma_a=\sigma_c$ and $\sigma_a=\sigma_c+1$ we can construct them easily but I haven't found any example for $\sigma_a=\sigma_c+\alpha$. Thank you.