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The notion of analytic conductor of a generic representation of $\mathrm{GL}(n)$ has been defined by Iwaniec and Sarnak, and since then is at the heart of many works in analytic number theory and used for different purposes. I am interested in understanding the consistency between the different definitions arising in the literature, and how much freedom we have in its definition.

Conrey et al. define it from the functional equation written in the following form (assuming every representation is self-dual for convenience) $$L(s, \pi) = \varepsilon_\pi X(s, \pi) L(1-s, \pi),$$

and then set that the (log-)conductor of $\pi$ is $$c_1(\pi) = |X'(1/2, \pi)|.$$

Indeed, since $|X(1/2,\pi)|$ takes value one, it appears as measure the defect of $L(s, \pi)$ to be symmetric (up to the sign), and thus a notion of complexity of $\pi$.

On the other side, other authors (Iwaniec and Sarnak, Michel, Kowalski, etc.) introduce the more explicit form of the $\gamma$-factor $$\gamma(s, \pi) = Q_\pi^s \prod_i \Gamma_i(s+\mu_\pi(i)),$$

where $Q_\pi$ is the usual arithmetic conductor of $\pi$ and $i$ varies among archimedean places, $\Gamma_i$ being the usual slight modification of the gamma function associated to it, that is to say $\Gamma_\mathbf{R}(s) = \pi^{-s/2}\Gamma(s/2)$ for real places, and $\Gamma_{\mathbf{R}}(s)\Gamma_{\mathbf{R}}(s+1)$ at complex places. It is the factor that allows to complete the $L$-function in a symmetric one (always up to the sign). The analytic conductor is then introduced as $$c_2(\pi) = Q_\pi \prod_i (1+\mu_\pi(i)).$$

I am concerned about the consistency of these two definitions, apparently very similar. The second one can appear as more ad hoc and unadvised, yet is more common and Conrey et al. gave the other formulation in a general attempt to encapsulate analytic properties of L-functions.

To what extend are these two definitions consistent one with each other (I find they are not, yet up to some approximation of the digamma function by the logarithm, they are)?

And behind this computational question, a more philosophical one should be: to what extent can we afford to manipulate the notion of analytic conductor? (of course, this is dependent on the type of question addressed with this notion, so sometimes it is affordable to diverge by an additive constant, sometimes by a multiplicative factor, ...)

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In all cases, the analytic conductor is defined as the usual arithmetic conductor $q_{\pi}$ times a product of terms coming from the archimedean places. The issue is the definition of the terms at the archimedean places. It is important to note that the analytic conductor is, in practice, a working definition: as we essentially only ever care about asymptotics involving the analytic conductor, rather than exact formulae, we allow small shifts at the archimedean places that may change the definition slightly. Often one is only interested in the logarithm of the analytic conductor, in which case the distinction between the different definitions is minute.

In any case, you should think of the Iwaniec-Sarnak definition as being an approximation of the Conrey et al. definition: Conrey et al.'s definition involves digamma functions, whereas Iwaniec-Sarnak essentially uses Stirling's formula to approximate these digamma functions by their leading order term (with some artificial shifts to ensure positivity). As you can see in their paper, Conrey et al.'s definition seems to be more "accurate" in a particular way, but it's not without its issues (for example, I vaguely recall Andy Booker mentioning that there are issues with using it in trace formulae since it doesn't quite satisfy the conditions needed for the test function to be valid).

For example, Iwaniec-Kowalski (equation (5.6)) define the archimedean contribution to be \[\mathfrak{q}_{\infty}(\pi,s) := \prod_{j = 1}^{n} (|s + \kappa_j| + 3)\] for an $L$-function whose associated gamma factors are of the form \[\pi^{-\frac{ns}{2}} \prod_{j = 1}^{n} \Gamma\left(\frac{s + \kappa_j}{2}\right),\] where the local parameters $\kappa_j$ are assumed to satisfy $\Re(\kappa_j) > -1$.

The reason for the unnatural presence of the term $3$ is essentially to ensure that $\mathfrak{q}_{\infty}(\pi,s) > 1$ (or even $> e$, so that $\log \mathfrak{q}_{\infty}(\pi,s) > 1$). Again, this is for practical reasons, where we usually are interested in asymptotics involving the analytic conductor tending to infinity, so we must ensure that it doesn't instead tend to zero for silly reasons! (I believe this may be an issue with Conrey et al.'s definition if you are working with highly nontempered automorphic representations.)

There are some issues with this definition. For example, it isn't exactly functorial: you'd like a definition to make sense over any number field, but then if you do this for an $L$-function that has complex places, then the gamma factors look different, and the Legendre duplication formula gives a (slightly) different definition of the analytic conductor than you might expect. (See, for example, Appendix A and in particular Remark A.8 of this paper.)

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