### Browse L-functions associated to newforms $f$ on Hecke congruence groups $\Gamma_0(N)$ with trivial character:

These L-functions $L(s,f) = \sum a_n n^{-s}$ have an Euler product of the form $L(s,f)= \prod_{p\mid N} \left(1 - a_p p^{-s} \right)^{-1}\prod_{p\nmid N} \left(1 - a_p p^{-s} + p^{-2s} \right)^{-1}$ and satisfy the functional equation of the form $$\Lambda(s,f) = N^{s/2} \Gamma_{\mathbb{C} } \left(s + \frac{k-1}{2} \right)\cdot L(s, f) = \varepsilon \Lambda(1-s,f),$$ where $N$ is the level, $k$ is the weight and $a_n n^{\frac{k-1}{2} }$ are algebraic integers. Here, $\varepsilon = i^k \varepsilon_N$, where $\varepsilon_N$ is the Fricke eigenvalue of $f$, and so is equal to either $+1$ or $-1$ since these L-functions are associated to the trivial character. It is therefore called the sign of the L-function.

In the plot below, the L-functions are organized by the ordered pair $(N,k)$. For a given $(N,k)$, the color indicates the sign of the functional equation, and the horizontal grouping indicates the degree of the field containing the arithmetically normalized coefficients. See the legend for more details.