Contour Plot 3

Edit and compile if you like:

\documentclass[tikz]{standalone}

\usepackage{mathtools}

\let\Im\relax
\DeclareMathOperator{\Im}{Im}
\let\Re\relax
\DeclareMathOperator{\Re}{Re}

\usetikzlibrary{decorations.markings,positioning}

\providecommand{\poles}{
  \node (poles) at (2.5,1.5) {poles of $h(p_0)$};
  \draw[fill]
  (1.5,3) coordinate [circle,fill,inner sep=1pt,label=right:$p_1$] (p1)
  (2,-2) coordinate [circle,fill,inner sep=1pt,label=below:$p_2$] (p2)
  (-3,1) coordinate [circle,fill,inner sep=1pt,label=above:$p_3$] (p3)
  (-2,-1.5) coordinate [circle,fill,inner sep=1pt,label=above:$p_4$] (p4);
  \draw[ultra thin,gray] (poles) -- (p1) (poles) -- (p2) (poles.west) -- (p3) (poles) -- (p4);
}

\def\xr{3.5}
\def\yr{3}

\begin{document}
\begin{tikzpicture}[thick]

  % Axes
  \draw[->] (-\xr-0.4,0) -- (\xr+0.4,0) node [above left] {$\Re(p_0)$};
  \draw[->] (0,-\yr-0.7) -- (0,\yr+0.7) node[below left=0.1] {$\Im(p_0)$};

  % Matsubara frequencies
  \foreach \n in {-\yr,...,-1,1,2,...,\yr}{%
      \draw[fill] (0,\n) circle (1pt) node [right=0.1,font=\footnotesize] {$i \mkern2mu \omega_{_{\n}}$};}
  \draw[fill] (0,0) circle (1pt) node [above right=0.1] {0};

  % Right contour line
  \draw[xshift=5,blue!60!black,decoration={markings,mark=between positions 0.1 and 1 step 0.25 with \arrow{>}},postaction={decorate}] (0,-\yr-0.75) node [above right] {$C_1$} -- (0,\yr+0.75) arc (90:-90:\yr+0.75);

  % Left contour line
  \draw[xshift=-5,blue!60!black,decoration={markings,mark=between positions 0.1 and 1 step 0.25 with \arrow{>}},postaction={decorate}] (0,\yr+0.75) -- (0,-\yr-0.75) node [above left] {$C_2$} arc (270:90:\yr+0.75);

  % Poles
  \poles

\end{tikzpicture}
\end{document}