In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula:


$\displaystyle{{\frak{q}}\_{\infty}(s)=\prod\_{j=1}^{d}\left(\vert s+\kappa\_{j}\vert+3\right)}$


Where does this $+3$ come from? Is it related to the abscissa of convergence? If yes, how?



What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the analytic conductor?
There is nothing special about the value "3", it is there to ensure that $\log{\frak{q}}\_{\infty}(s)>0$ when $s\rightarrow\infty$.


