Through my years in nuclear engineering, it has always been the case that in physical relations, the arguments of transcendental functions, e.g., the exponential in the law of radioactive decay, $N=N_0 e^{-\lambda t}$, should be dimensionless so that the outcome is correspondingly dimensionless.
In purely empirical relations, however, I have observed that it's okay for transcendental functions to have dimensional arguments as long as they fit the results of experiments. For example, the energy spectrum of prompt fission neutrons is fitted by: $$\chi(E)=0.453 e^{-1.036E}\text{sinh}\sqrt{2.29E}.$$ Or does the 1.036 in the exponential have units of ${\text{eV}}^{-1}$?
Is my observation correct? And if so, what is the basis of all that? I tried to dig into dimensional analysis, but I couldn't understand much.
Update on Jan 14, 2020:
I've found an insight on Wikipedia that I think will, hopefully, enrich the discussion.
A problem with transcendental functions is that: applying a non-algebraic operation to a dimensional quantity creates paradoxical results. For instance, $\text{log}_a(5L)=\text{log}_a(3)+\text{log}_a(L)$, where $L$ is the dimension of length, and $a$ is an arbitrary base.
This also raises the question: is $\text{log}_a(L)$ dimensional? Does a transcendental function become dimensional when fed with a dimensional argument?