On extension to the left halfplane of the scalar product of Hecke 𝐿-series with magnitude characters

Distribution of values of Fourier coefficients for modular forms of weight 1

“…The way of proving Theorem 10 is related to the efforts made by a number of mathematicians to solve Linnik's problem concerning the scalar product of the Hecke L-functions of algebraic numbers fields (see [4][5][6][7][8]). * The methods of these authors differ from the method of "Rankin convolution," which was also used in [9] in order to study Linnik's problem.…”

“…XI X2 (33) where ~• (j = 1, 2) denotes summation over the characters of the ideal class group of k. It follows from the results of [4][5][6][7][8] that the scalar products in (33) can be expressed via the Hecke L-functions, Lk(s,x,) * Lk(s, xz) = Lk(s'x<'))Lk(s'x(2))…”

The spatial distribution of algebraic integers on constant‐norm hypersurfaces

“…Then it is classically well-known that we can find L, U e ^ such that 3A. Scalar products ofHecke L-functions (see [4,5,9,14]). Let K, L be arbitrary algebraic number fields, Following Moroz [10] we obtain a continuation of £(s, x, </0 into the region a>\, as follows.…”

Scalar product of L-functions with Grössencharacters: its meromorphic continuation and natural boundary.

“…It is reminiscent of the P. DraxFs theorem ( [4], Satz 2) concerning the structure of the scalar product (2) in C*; this theorem follows 4 ) See also [32], p. 157-159. In these papers (see, [21], [22], [23]) some arithmetical applications of the function (1) are given; further arithmetical applications can be found in the work of A. I. Vinogradov, [33], where this function has been continued to the half-plane Res>i by an argument close to the one used here, in the proof of lemma 9. The corollary l is a more precise Statement of our previous results (see [25]); it should be compared to the discussion of the two quadratic fields case by O. M. Fomenko, [5], and E. Gaigalas, [6].…”

“…Statement of the main theorem and the plan of the exposition Let k be an algebraic number field (that is, a fmite extension of the field 0 of rational numbers), and k i be a finite extension of k; we set (k.:k)^d i and assume that d t Ξ> d 2 ^ * * · jj> rf r , l i» i f Si r. Let χ. be a Gr ssencharacter in fc . ., L kr (s 9 χ,) over k is defmed (e.g., [4], [19], [21] and [33]) by an absolutely convergent for Res > l series: Given any Gr ssencharakter t/r in an algebraic number field M one defines a Hecke fupction L M (s, ψ) associated with ψ by an absolutely convergent for Re^> l series where Hl runs over all the integral ideals in M. This function is known (e.g., [3], [9]) to have a holomorphic continuation to the whole complex plane C (except for a possible pole at je l, when φ is a trivial character), The scalar product (or convolution) of the Hecke functions L kl (s, ^),.…”