On the precise form of $\mathrm{GL}(3)$ (and others) L-functions

Question

$\DeclareMathOperator\GL{GL}$Typical $\GL(3)$ automorphic L-functions are often depicted as looking like the following Dirichlet series: $$L(s, \pi) = \sum_{n >0} \frac{A(1,n)}{n^s}$$

Is there a specific reason for the L-function to be of this form (rather than with $A(n,1)$ say, or $A(n,n)$ or $A(n,m)$?). I would like to see a link with the classical cases of the zeta function or of $\GL(2)$, where it comes from the Fourier expansion and defining the L-function as a Mellin transform of the form itself.

Answer 1

Whether you use $A(1,n)$ or $A(n,1)$ is just a convention. If one sequence defines $L(s,\pi)$, then the other one defines the dual $L$-function $L(s,\tilde\pi)$. For example, we could define the Dirichlet $L$-function $L(s,\chi)$ as $\sum_n \overline{\chi(n)} n^{-s}$, it would not change a thing. BTW in Goldfeld's $\mathrm{GL}_n(\mathbb{R})$ book what he calls $L(s,\pi)$ for $n=3$ is really $L(s,\tilde\pi)$, i.e. the definition he gives for $n=3$ is not compatible with the definition he gives later for general $n$. If I recall correctly.

The coefficients $A(n,n)$ or $A(n,m)$ would not produce nice Euler factors (among other things). That is, these coefficients don't satisfy as nice multiplicative-recursive properties as $A(1,n)$ or $A(n,1)$. I recommend Goldfeld-Hundley's second volume for the general definition of $L(s,\pi)$. I also recommend the excellent lecture notes (in various books) by Cogdell. Finally, you might want to read my post here as it summarizes the remarkable paper of Kondo-Yasuda. The point is that one can define the Euler factors in terms of Hecke eigenvalues even at the ramified places (extending the classical work of Tamagawa and Satake at the unramified places).

In the end the answer boils down to mathematics being a mixture of science and art. Good definitions are those that work.

For the classical theory of $\mathrm{GL}_1$, see e.g. Weil: Basic Number Theory or Neukirch: Algebraic Number Theory. For the classical theory of $\mathrm{GL}_2$, any introductory textbook on automorphic forms will do (e.g. Goldfeld-Hundley first volume, or Bump, or books by Iwaniec).