I know that we can analytically continue $L$-series by taking the Mellin transform of a sutiable theta function. Technically we need a transformation law for the theta function at $0$ and $\infty$ as well and then we need to symmetrize the result as much as possible. In any case, for say $\zeta(s)$, a Diricihlet $L$-series $L(s,\chi)$, and the $L$-series $L(s,f)$ for a level $1$ modular form $f$, we have the following correspondences:
\begin{align*} \zeta(s) &\longleftrightarrow \omega(x) = \sum_{n \ge 1}e^{-\pi n^{2}x} \longleftrightarrow \vartheta(s) = \sum_{n \in \mathbb{Z}}e^{-\pi n^{2}s} \\ L(s,\chi) &\longleftrightarrow \omega_{\chi}(x) = \sum_{n \ge 1}\chi(n)n^{\delta_{\chi}}e^{-\pi n^{2}x} \longleftrightarrow \vartheta_{\chi}(s) = \sum_{n \in \mathbb{Z}}\chi(n)n^{\delta_{\chi}}e^{-\pi n^{2}s} \\ L(s,f) &\longleftrightarrow f(iy) = \sum_{n \ge 1}a(n)e^{-\pi ny}, \end{align*} here $\delta_{\chi} = 0,1$ depending on if $\chi$ is even or odd and $a(n)$ is the $n$-th Fourier coefficient of $f$.
I'm curious if there is good intution for why the first two theta functions $\omega$ and $\omega_{\chi}$ are the right ones to look at (beyond the change of variables trick Riemann did with $\Gamma\left(\frac{s}{2}\right)$ for $\zeta(s)$ and then augmenting the argument for $L(s,\chi)$). For a modular form, $f(iy)$ seems like the natural thing to consider since it will immeditely have a transformation law coming from the modularity of $f$ and the coefficients of the series literally match thoes of the $L$-series. This is also the same for $\omega$ but not for $\omega_{\chi}$ if $\chi$ is odd. We also have the additional issue of $n^{2}$ in the exponenet rather than $n$. If I had to guess, the diference in $n^{2}$ vs $n$ comes from that $L(s,f)$ has a degree $2$ Euler product so there will be two local factors at infinity and so the Gamma function you get out of the Mellin transform must be of the form $\Gamma(s)$ rather than $\Gamma\left(\frac{s}{2}\right)$. So you can guess this bit from the Euler product.
For more general $L$-series than those I've mentioned above is there a way to "know" what the theta function will be? Or is this part of the difficult work + experience?