LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £ -FUNCTIONS 201- Hoffstein/ Lockhart proved the analog of Siegel's theorem for the sym-
metric square £ -functions L(sym^2 n , s ), for n E Ag(Q) [HL], and shortly
afterwards Goldfeld, Hoffstein and Liemann [GHL] eliminated the ex-
ceptional zero effectively of the adjoint square lift £-function, L(Adn, s ),
for non-dihedral n. (When n is dihedral, L(Adn, s ) is divisible by the£-
function of a quadratic character, and then Siegel's theorem is available.) - Hoffstein/ Ramakrishnan proved that there is no exceptional zero for
L(n, s), n E Ag(Q) and for many n of degree 3 [HR]; then Banks [Ban]
completed their argument and so there is no exceptional zero for L (n , s ),
n E A~(Q). - Ramakrishnan/ Wang [RaW] proved that L(n ® n', s ) and L(sym^2 n ®
sym^2 n , s ) for n , n' E Ag(Q) have no exceptional zero, excepted when the
corresponding £-functions are divisible by the £-functions of quadratic
characters. In these latter cases (which are completely characterized), an
exceptional zero, if it ever occurs, comes only from the quadratic charac-
ter £-functions (so that one may apply Siegel's theorem).
Again, the method to eliminate the exceptional zero of some L ( s) is to construct a
Dirichlet series D(s ) with non-negative coefficients that is divisible by L(s ) to an
order strictly larger than the order of the pole of D(s ) at s = 1, and to then apply
Lemma 1.2.1. The construction of D(s ) depends on the recent progress made in
the direction of the Langlands functoriality conjecture. For instance, in the case of
L(sym^2 n , s ) for a self-dual n E Ag(Q), Goldfeld, Hoffstein and Liemann used
D(s ) = ( (s )L(sym^2 n , s )^2 L(sym^2 n ® sym^2 n , s )
= ( (s )L(sym^2 n , s )^3 L(sym^2 (sym^2 n), sand is was proved by Bump/ Ginzburg (and it follows from the recent results of
Kim/ Shahidi) that the last factor has a simple pole at s = 1.
More generally, Hoffstein/ Ramakrishnan [HR] proved that:
Theorem 1.6. If the Functoriality Conjecture for Pairs is true then principal
L -functions of degree d > 1 have no exceptional zero at all!
Proof. Given n E A~(Q) with d > 1 and self-dual. The functoriality conjecture for
pairs implies that n ~ n has a (cuspidal) constituent r =/=-1, n. Setting II = 1 EEi r EEi n
and D(s) = L (II ®fr , s ), one sees that D(s ) has a pole of order 3 at s = 1 and is
divisible by L(n, s)^2 L(n EEi r , s)L(n EEi i , s ); the conjecture again implies that n is a
constituent of n EEi r and n EEi i (by Remark 1.2). So D ( s ) is divisible by L ( n , s )^4 and
one concludes by Lemma 1.2.1. D
For a more complete account on Landau/ Siegel zeros and for some methods to
attack the problem, we refer to the surveys [IS3, Ra2].
1.3. Bounds for L-functions on the critical line
An important consequence of GRH is the Generalized Lindelof Hypothesis, which
predicts a sharp upper bound for the size of L( n , s) when s is on the critical line:
Generalized Lindelof Hypothesis (GLH). For any c; > 0, and ~es = 1/ 2, one has
IL(n , s)I « Q7r(t) ", the implied constant depending on c;.
In full generality; the best one can prove (unconditionally) is the following: