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, ...)