Browse \(L\)-functions associated to newforms \(f\)

For a newform $f\in S_k^{\rm new}(N,\chi)$ with q-expansion $f(z)=\sum a_n n^{(k-1)/2}q^n$, where $q=\exp(2\pi i z)$, the $L$-function \(L(s,f) = \sum a_n n^{-s} \) has an Euler product of the form \[ L(s,f)= \prod_{p\mid N} \left(1 - a_p p^{-s} \right)^{-1}\prod_{p\nmid N} \left(1 - a_p p^{-s} + \chi(p) \cdot p^{-2s} \right)^{-1} \] and satisfies the functional equation \[ \Lambda(s,f) = N^{s/2} \Gamma_{\mathbb{C} } \left(s + \frac{k-1}{2} \right)\cdot L(s, f) = \varepsilon \overline{\Lambda}(1-s,f), \] where $\varepsilon$ is the sign of the functional equation. When $\chi$ is the trivial character, $\varepsilon=\pm 1$, and in general it is a root of unity.


In the plot below, each dot corresponds to a primitive \(L\)-function of an embedded cusp form. These are organized:
  • analytic conductor, in the $x$-axis;
  • imaginary part of the first postive nontrivial zero on the critical line, in the $y$-axis;
  • weight, the color of each dot.