I asked this question on m.SE in an attempt to find out the right words to say for these questions I am about to ask.
In his great answer, Matthew Emerton explained that (cuspidal) automorphic L-functions correspond to the Dirichlet series with "nice" properties like having a reflection equation, a meromorphic continuation to the entire complex plane and a suitable analogue of the Riemann hypothesis. This leads me to my first question:
1) Are there Dirichlet series that cannot be classified as (cuspidal) automorphic L-functions, yet still possess a critical line of nontrivial zeroes?
Now, I come to the question I had been meaning to ask here. It is known that functions like Riemann $\zeta$ and the Ramanujan Dirichlet series admit a "Riemann-Siegel" decomposition; that is, letting $\sigma$ denote the position of the "critical line" of the Dirichlet series $g(s)$, they can be expressed as
where $z(t)$ and $\vartheta(t)$ are "Riemann-Siegel" functions corresponding to the Dirichlet series $g(s)$. The value of $z(t)$ is that it eases the task of finding nontrivial zeroes of the corresponding Dirichlet series (essentially helping to verify its corresponding "hypothesis"). My question now is
2) Do all (cuspidal) automorphic L-functions have a "Riemann-Siegel" decomposition? If not, what restrictions are there for them to possess such a decomposition?
My motivation is more of curiosity than anything else. Hopefully this is not too elementary a question!