I was reading the paper "Integral moments of $L$-functions" by Conrey et al. They have used the following sharp cutoff version of the approximate functional equation for "Selberg class" $L$-functions.
\begin{equation*} L(s)=\sum_{m<x}\frac{a_m}{m^s}+\varepsilon X_L(s)\sum_{n<y} \frac{\bar{a}_n}{n^{1-s}}+\text{remainder}, \end{equation*}
where $X_L(s)=\frac{\bar{\gamma}_L(1-s)}{\gamma_L(s)}$, $\gamma_L(s)$ is the gamma factor, and $\varepsilon$ is the sign of the $L$-functions.
I don't find any standard reference for this result. What would be the remainder of this of the above equation? Do they depend on the degree of the $L-functions?
This is purely a heuristic. It may be reasonable due to the rough fact that approximation functional equations are typically of the form $$ L(s, f) = \sum_{m} \frac{a(m)}{m^s} V_s(m/X\sqrt{q}) +\varepsilon(f, s)\sum_m \frac{\overline{a(n)}}{n^{1 - s}} V_{1-s}(nX / \sqrt{q}) + \mathrm{MT},$$ where $\mathrm{MT}$ is a "main term" coming from poles (if they exist), and $V$ is a rapidly decaying cutoff function. It's possible to write down other approximation functional equations with slightly different forms, but this doesn't affect the following heuristic arguments.
Heuristically each cutoff function is initially constant and then rapidly decays to $0$, so one might hope that replacing each sum by a finite sum of an appropriate length with no weighting would indicate what's going on. The paper you mention carries out these heuristic arguments and shows that they seem to agree with known cases and accurately predict unproved cases.
There is a different motivation for approximate functional equations of this shape. The Riemann Siegel formula for the Riemann zeta function is $$ \zeta(s) = \sum_{n \leq N} \frac{1}{n^s} + \varepsilon(s) \sum_{m \leq M} \frac{1}{m^{1-s}} + R(s) $$ for a remainder $R(s)$ that is explicit but which I don't write down.
This is of the desired form, but the derivation relies on the fact that the coefficients are totally multiplicative. A similar expansion is attainable for Dirichlet $L$-functions (see for example An Approximate Functional Equation for Dirichlet L-Functions by D. Davies, in Proc. Roy. Soc. London Series A, 1965). For $L$-functions of degree greater than $1$, it seems that smoothing is necessary.