Do quadratic curves have L-functions?

The genus degree formula for a smooth projective plane curve of degree $d$ is $$g = \frac{(d-1)(d-2)}{2}$$ so degree 2 gives $g = 0$, and degree 3 gives $g = 1$.

For a smooth projective curve $C$ of genus $g$ over $\mathbb{Q}$, the hasse-weil zeta function can be factored into $$Z(C, s) = \frac{\zeta(s)\zeta(s-1)}{L(H^1(C), s)}$$ where $\zeta(s)$ is the normal riemann zeta, and $L(H^1(C), s)$ is the L function corresponding to the first etale cohomology group $H^1$ (with dimension 2g).

For $g=1$, $H^1$ is 2D, so $L(H^1, s)$ is a nontrivial Euler product (the normal L function you work with, modularity theorem, etc etc blah blah)

For $g=0$, $H^1$ is 0D. So $L(H^1, s) = 1$ and the hasse-weil zeta instead is just $$Z(C, s) = \zeta(s)\zeta(s-1)$$ (up to corrections at finitely many primes of bad reduction). So they do ""have"" an L function, but it is trivial/nothing interesting comes out of it. Their connection to modular forms exists only in the sense that dirichlet characters appear in its eisenstein series.

there's probably better views like from langlands POV or somethin but the gist is basically their L function is trivial and the structure is incredibly well understood by this poijt