L-functions of degree 1 are associated to Dirichlet characters and they are called
Dirichlet L-functions.
These L-functions have a functional equation of the form
\[
\Lambda(s,\chi) = q^{s/2} \Gamma_{\R} (s+a) L(s,\chi) = \varepsilon \Lambda(1-s,\overline{\chi}),
\]
where $q$ is the conductor of $\chi$, $\varepsilon$ is the sign and $a$ is 1 (or 0) when $\chi$ is odd (or even).
In particular, the only Dirichlet L-function with conductor 1
is the Riemann zeta function.
Degree 1 L-functions can also be obtained from Artin representations of dimension 1.

The plot below shows the primitive characters
arranged by the conductor and the
order of the character,
with color indicating the parity (blue is even) of the
Dirichlet character. Clicking on an entry will
take you to the associated Dirichlet L-function.