## Algebra & Number Theory

### The behavior of Hecke $L$-functions of real quadratic fields at $s=0$

#### Abstract

For a family of real quadratic fields ${Kn=ℚ(f(n))}n∈ℕ$, a Dirichlet character $χ$ modulo $q$, and prescribed ideals ${bn⊂Kn}$, we investigate the linear behavior of the special value of the partial Hecke $L$-function $LKn(s,χn:=χ∘NKn,bn)$ at $s=0$. We show that for $n=qk+r$, $LKn(0,χn,bn)$ can be written as

$1 1 2 q 2 ( A χ ( r ) + k B χ ( r ) ) ,$

where $Aχ(r),Bχ(r)∈ℤ[χ(1),χ(2),…,χ(q)]$ if a certain condition on $bn$ in terms of its continued fraction is satisfied. Furthermore, we write $Aχ(r)$ and $Bχ(r)$ explicitly using values of the Bernoulli polynomials. We describe how the linearity is used in solving the class number one problem for some families and recover the proofs in some cases.

#### Article information

Source
Algebra Number Theory, Volume 5, Number 8 (2011), 1001-1026.

Dates
Revised: 24 March 2011
Accepted: 8 May 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729728

Digital Object Identifier
doi:10.2140/ant.2011.5.1001

Mathematical Reviews number (MathSciNet)
MR2948469

Zentralblatt MATH identifier
1260.11066

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$

#### Citation

Jun, Byungheup; Lee, Jungyun. The behavior of Hecke $L$-functions of real quadratic fields at $s=0$. Algebra Number Theory 5 (2011), no. 8, 1001--1026. doi:10.2140/ant.2011.5.1001. https://projecteuclid.org/euclid.ant/1513729728

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