Hecke L-functions Research Articles

Heckel Plots as Indicators of Elastic Properties of Pharmaceuticals

In this study the usefulness of different kind of Heckel treatments to describe the elastic properties of the compressed materials was evaluated. Two methods were used to find the values for Heckel function, namely, tablet-in-die method and ejected tablets method. The slope values were calculated by means of the first derivative of the Heckel function. The reliable conclusions from the total elastic recovery were possible to obtain using the reciprocal of the difference of the Heckel upward plot slopes obtained from the two methods used-in this study. The downward part of the Heckel plot was noticed to be useful for describing the fast elasticity during the tableting process.

Kummer's criterion for the special values of HeckeL-functions of imaginary quadratic fields and congruences among cusp forms

Here, arrow (i) refers to our previous papers [9, 12] and [13], in which we have shown that a fixed primitive cusp form f of Sk(F I(N)) has congruences with other cusp forms of Sk(F~(N)) modulo the special value at s = k of a certain zeta function off . In this paper, we will construct the second arrow for the cusp form f associated with a Hecke character of an imaginary quadratic field K, and consequently obtain the third. Namely, in w167 6 we will show, as in the works of Coates-Wiles [3] and Robert [23], that the split primes of K which divide this special value are irregular in an appropriate sense. In this case, the zeta function o f f mentioned above is a Hecke L-function of K. Such a criterion of irregularity for Hurwitz numbers has already been obtained in [3] and [233. However, our result together with those of [3] and [23] does not cover all the L-values of Hecke L-functions of K whose algebraicity was given by Damerell [5] (for details, see below). Our method is analogous to that of Ribet [21] (see also Wiles [35]), who used congruences between cusp forms and Eisenstein series to prove the non-vanishing of certain eigenspaces (relative to the action of Gal(Q/Q)) of the p-primary part of the class group of the field of p-th roots of unity. By adopting this approach, we will obtain such an information of eigenspaces again in our case (see Theorem 0.1 below). To be more specific, let 2 be a Hecke character of K satisfying

An estimate of the Hecke L-functions on the critical line

One obtains, under certain restrictions, an estimate for the Hecke L -functions on the critical line, similar to the estimate for the Riemann zeta-function.

KRONECKER'S LIMIT FORMULA IN A REAL QUADRATIC FIELD

In this paper the author obtains a formula representing the free term of the Laurent expansion of the zeta functions of absolute and ray classes of a real quadratic field at the point 1, as well as the value of Hecke's L-function corresponding to the signature character. The formula contains Dedekind sums and sums analogous to them in which functions depending on the same arguments as in ordinary Dedekind sums appear. Bibliography: 4 titles.

On the convolution of Hecke L ‐functions

§1. Let k be an algebraic number field of finite degree over the field Q of rational numbers and Ki be an extension of k of degree (Ki:k) = ni, i = 1,2. We choose a Hecke Grossencharakter Xi in Ki and consider the L-functionassociated with Xi see [1, 2]. It is known to be a meromorphic function on the whole complex plane. We are interested here in the properties of the convolution of functions LK1, LK2 over k defined bywhereand a (and n) run over integral ideals of Ki (and k), whose norm NKi/k a is equal to n. The function (2) is sometimes called the “scalar product of the Hecke L-functions” «3—11». The object of the paper is the following theorem.

The non-vanishing of certain Hecke L-functions at the center of the critical strip

The non-vanishing of certain Hecke L-functions at the center of the critical strip

On computing Artin 𝐿-functions in the critical strip

This paper gives a method for computing values of certain nonabelian Artin L-functions in the complex plane. These Artin L-functions are attached to irreducible characters of degree 2 of Galois groups of certain normal extensions K of Q. These fields K are the ones for which G = Gal ( K / Q ) G = {\operatorname {Gal}}(K/{\mathbf {Q}}) has an abelian subgroup A of index 2, whose fixed field Q ( d ) {\mathbf {Q}}(\sqrt d ) is complex, and such that there is a σ ∈ G − A \sigma \in G - A for which σ a σ − 1 = a − 1 \sigma a{\sigma ^{ - 1}} = {a^{ - 1}} for all a ∈ A a \in A . The key property proved here is that these particular Artin L-functions are Hecke (abelian) L-functions attached to ring class characters of the imaginary quadratic field Q ( d ) {\mathbf {Q}}(\sqrt d ) and, therefore, can be expressed as linear combinations of Epstein zeta functions of positive definite binary quadratic forms. Such Epstein zeta functions have rapidly convergent expansions in terms of incomplete gamma functions. In the special case K = Q ( − 3 , a 1 / 3 ) K = {\mathbf {Q}}(\sqrt { - 3} ,{a^{1/3}}) , where a > 0 a > 0 is cube-free, the Artin L-function attached to the unique irreducible character of degree 2 of Gal ( K / Q ) ≅ S 3 {\operatorname {Gal}}(K/{\mathbf {Q}}) \cong {S_3} is the quotient of the Dedekind zeta function of the pure cubic field L = Q ( a 1 / 3 ) L = {\mathbf {Q}}({a^{1/3}}) by the Riemann zeta function. For functions of this latter form, representations as linear combinations of Epstein zeta functions were worked out by Dedekind in 1879. For a = 2 , 3 , 6 a = 2,3,6 and 12, such representations are used to show that all of the zeroes ρ = σ + i t \rho = \sigma + it of these L-functions with 0 > σ > 1 0 > \sigma > 1 and | t | ⩽ 15 |t| \leqslant 15 are simple and lie on the critical line σ = 1 / 2 \sigma = 1/2 . These methods currently cannot be used to compute values of L-functions with Im ⁡ ( s ) \operatorname {Im} (s) much larger than 15, but approaches to overcome these deficiencies are discussed in the final section.

On the values of Hecke's L-functions at non-positive integers

On the values of Hecke's L-functions at non-positive integers

Approximate functional equation for Hecke's L-functions of quadratic field

Approximate functional equation for Hecke's L-functions of quadratic field

The parity of the rank of the mordell-weil group

L(s) = i(s)% l)/&(s). Here c(s) is the Riemann zeta-function, and the series L,(s) is convergent for Re(s) > $. It is conjectured that L,(s) may be continued analytically over the whole plane; certainly this is so when r has complex multiplication (in particular, when ab = 0), for then L,(s) is a Hecke L-function with Grossencharaktere (see [6]); it is also so for a much larger dynasty ofcurves founded by Eichler [7] and Shimura [IO]. As is well known, the points of IY form a group; by the Mordell-Weil theorem, the subgroup d consisting of the rational points of r is finitely generated. Write g for the number of independent generators of infinite order of &, and write y for the order of the zero of L,(s) at s = 1 (so y = 0 if&(l) # 0). This makes sense when&.(s) can be continued-from now on we restrict ourselves to such cases. It has been conjectured [l, 2,4] that

On the zeros of Hecke's L-functions III

On the zeros of Hecke's L-functions III

On the zeros of Hecke's L-functions II

On the zeros of Hecke's L-functions II

On The zeros of Hecke's L-functions I

On The zeros of Hecke's L-functions I