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\begin{document}

If $K$ is a $p$-adic field, any character $c_0:U_K \longrightarrow T$ will
factor through a finite quotient. $\{1 + p^n\}$ form a fundamental system of
neighborhoods in $U_K$.  Therefore, there is an integer $n \geq 0$ such that
$c_0\mid_{1 + p^n} \;\equiv 1$.
\medskip

If $n$ is the smallest nonnegative integer such that $c_0\mid_{1 + p^n}
\;\equiv 1$,
we define the conductor of the quasicharacter $c = c_0| \; |^s$ to be
$\mathcal{F}(c_0) = p^n \subseteq \ok{K}$.
\medskip

Note that $c$ is unramified if and only if $\mathcal{F}(c_0) = \ok{K}$.
\bigskip

Recall that we have normalized the Haar measure $dx$ on $K$ as Lebesgue
measure
if $K = \R$, as twice Lebesgue measure when $K = \C$, and such that
$\int_{\ok{K}} dx = (N\msr{D})^{-1/2}$ for $p$-adic $K$.
\medskip

Now we normalize Haar measure $d^\times x$ on $K^\times$, by
$d^\times x = \frac{dx}{|x|}$ for archimedean $K$ and, for $K$ $p$-adic,
$d^\times x = \frac{Np}{Np - 1} \frac{dx}{|x|}$.
\bigskip

\section{The Local Zeta Function}

\begin{Def}
\label{D:IV.4.2}
The Schwarz-Bruhel functions on a local field $K$, denoted $\msr{S}(K)$, are
defined
as follows:

If $K = \R$, $f \in \msr{S}(K)$ if $f \in C^\infty$ and for every polynomial
$p$,
$$\sup_{x \in \R} |p(x)\frac{d^n f}{dx^n}| < \infty, \forall n \geq 0.$$

If $K = \C$, $f \in \msr{S}(K)$ if $f \in C^\infty$ and for every polynomial
$p$,
$$\sup_{x,y \in \R} |p(x,y)\frac{\partial^{m+n} f}{\partial x^m \partial
y^n}| < \infty,
\forall m, n \geq 0.$$

If $K$ is $p$-adic, $f \in \msr{S}(K)$ if $f$ is locally constant with
compact support.
\end{Def}
\bigskip

\begin{exa}
\begin{enumerate}
\item $e^{-x^2} \in \msr{S}(\R)$.
\item Any $C^\infty$ function with compact support is in $\msr{S}(\R)$.
\item The characteristic function of $x + p_n$ is in $\msr{S}(\Q_p)$.
\end{enumerate}
\end{exa}
\bigskip

\begin{Def}
\label{D:IV.4.1}
Let $K$ be a local field, $f \in \msr{S}(K)$.  Then the local zeta function
is defined as
$$\zeta(f, c) = \int_{K^\times} f(x) c(x) d^\times x ,$$ where
$d^\times x$ is Haar measure on $K^\times$ and $c$ is a quasicharacter of
$K$.
\end{Def}
\bigskip

\begin{thm}
\label{T:IV.4.1}
If $f \in \msr{S}(K)$, then $\hat{f} \in \msr{S}(\hat{K})$.
\end{thm}
\bigskip

\noindent {\bf{Notation.}}
Given a quasicharacter $c = c_0| \; |^s$, $\zeta(f,c)$ will sometimes be
denoted as
$\zeta(f,c_0,s)$ and $\re{s}$ as $\re{c}$.

\end{document}