LECTURE 1
Analytic properties of individual £-functions
1.1. Automorphic L-functions
1.1.1. Principal £-functions
Let 7r = ®7rp be an automorphic cuspidal (irreducible) representation of GLd(Aq)
with unitary central character (we denote by A~(Q) the set of all such represen-
tations). By the general theory (Hecke, Gelbart/Jacquet, Godement/Jacquet: see
Cogdell's lectures in this volume [Co2]), 7r admits an £-function:
II
L(7r, s) = Lp(1r, s) = "'°""' L.., An ~-( n)
p<oo n~lThis is an Euler product absolutely convergent for ~es sufficiently large where for
each (finite) prime p, the inverse of the local factor Lp(1r, s) is a polynomial in p-s
of degree ~ d:
d
Lp(7r,s)-l = L(7rp,s)-l = IJ(l-O'.n,is(p)).
i=l p
The £-function of 7r is completed by a local factor at the infinite place, given by a
product of d Gamma factors:
d
Loo(1r, s) = L(7r 00 , s) = IJrR(s - μn,i), I'R(s) = 7r-s/^2 r(s/2);
i=lthe coefficients {an,i(P)}i=l...d (resp. {μn,i h=i...d) will be called the local para-
meters of 7r at p (resp. oo). The completed £-function L 00 (7r, s)L(7r, s) has the
following analytic properties:
- L 00 (7r, s)L(7r, s) has a meromorphic continuation to the complex plane
with at most two simple poles; the latter occur only if d = 1 and 7r = I· lit
for some t E R, in which case L( 7r, s) = ( ( s + it) and the poles are at
s = -it, 1 - it.
- L 00 (7r, s )L(7r, s) satisfies a functional equation of the form
(1.1) q~1^2 L 00 (7r, s)L(7r, s) = w(7r)q~l-s)/^2 L 00 (ir,1 - s)L(ir, 1 - s),187