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

Typical$\DeclareMathOperator\GL{GL}$Typical $GL(3)$$\GL(3)$ automorphic L-functions are often depicted as looking like the following Dirichlet series: $$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)$$\GL(2)$, where it comes from the Fourier expansion and defining the L-function as a Mellin transform of the form itself.

On the precise form of $GL(3)$ (and others) L-functions

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.

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

$\DeclareMathOperator\GL{GL}$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.

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Automorphic
Automorphic
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On the precise form of $GL(3)$ (and others) L-functions

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.