### 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 \begin{equation} \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), \end{equation} 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.