I am interested in the connection between particular Dirichlet series' abscissa of convergence and the poles of L-functions.

Let $D(z) = \sum_{n=1}^\infty\frac{a_n}{n^z}$ be a Dirichlet series admitting meromorphic continuation, with abscissa of absolute convergence (resp. conditional convergence) $\sigma_a \in \mathbb{R}$ (resp. $\sigma_c$), and let $L(z)$ be the corresponding L-function to $D(z)$. I would like to know the connection between $\sigma_c$ and the poles of $L$.

It is known that if $a_n \geq 0 \space\forall n$, then $L$ will have a pole on the real axis at $\sigma_a$ (for example, Hardy/Riesz, General Theory of Dirichlet Series, Thm 10). It is also known that $\sigma_c \leq \sigma_a \leq \sigma_c +1$. I want to know what happens to $L$ when the second inequality is not strict equality.

It is obvious if $\sigma_c = \sigma_a - 1$, that $L$ cannot have any poles in the region $S = \{z \in \mathbb{C} \space \vert \space \sigma_a-1 <\Re(z)\leq \sigma_a \}$ (It can even happen that $L$ is entire in this case - for example, Dirichlet $\eta$ function).

Define $T = \{ \Re(z) \space \vert \space L \space \text{has a pole at z} \in \mathbb{C} \}$.

My question is as follows:

If $\sigma_c > \sigma_a - 1$, must it be the case that $\sup T = \sigma_c$? How can I show this?

If needed, assume $L$ has any combination of an Euler product, functional equation, or Ramanujan conjecture.

I have seen approaches using Perron's formula for particular functions, but this technique is far from general. Thanks for any input!