Goldfeld and Huang Res Math Sci (2018) 5:16
Super-positivity of a family of L-functions
in the level aspect
and Bingrong Huang
Department of Mathematics,
Columbia University, New York,
NY 10027, USA
Full list of author information is
available at the end of the article
An automorphic self dual L-function has the super-positivity property if all derivatives of
the completed L-function at the central point s = 1/2 are nonnegative and all
derivatives at a real point s > 1/2 are positive. In this paper, we prove that at least 12%
of L-functions associated to Hecke basis cusp forms of weight 2 and large prime level q
have the super-positivity property. It is also shown that at least 49% of such L-functions
have no real zeros on Re(s) > 0 except possibly at s = 1/2.
Keywords: Level, L-function, Molliﬁcation, Real zeros, Super-positivity, Zero-density
The notion of super-positivity of self dual L-functions was introduced in withbreak-
through applications to the function ﬁeld analog of the Gross–Zagier formula for higher
derivatives of L-functions. We say a self dual L-function has the super-positivity property
if all derivatives of its completed L-function (including Gamma factors and power of the
conductor) at the central value s = 1/2 are nonnegative, all derivatives at a real point
s > 1/2arepositive,andifthek
th derivative is positive then all (k
+ 2j)th derivatives
are positive, for all j ∈ N.See for more details.
In this paper, we continue our investigation of super-positivity of L-functions associated
to classical modular forms as begun in . Let q be a ﬁxed large prime. Let S
space of holomorphic weight 2 cusp forms of level q,andH
(q) that are
eigenfunctions of all the Hecke operators and have the ﬁrst Fourier coeﬃcient a
(1) = 1.
Our main theorem in this paper is as follows.
Theorem 1.1 There are inﬁnitely many modular forms f of weight 2 and prime level such
that L(s, f ) has the super-positivity property. In fact, the proportion of f ∈ H
have the super-positivity property is ≥ 12%, when q is a suﬃciently large prime.
In the course of proving Theorem 1.1, we also obtained the following result about real
zeros of L-functions as a by-product.
Theorem 1.2 There are inﬁnitely many modular forms f of weight 2 and prime level such
that L(s, f ) has no nontrivial real zeros except possibly at s = 1/2. In fact, the proportion of
f ∈ H
(q) with this property is ≥ 49%, when q is a suﬃciently large prime.