Quadratic twists

Given a newform $f(z)=\sum_{n=1}^\infty a_n e(nz)$ of weight $k$ and level $N$ we can form its $L$-function

$$L_f(s)=\sum_{n=1}^\infty \frac{a_n/n^{(k-1)/2}}{n^s}.$$

This $L$-function satisfies a functional equation under $s\to 1-s$ so that the $s=1/2$ is the center of the critical strip.

Let $d$ be a fundamental discriminant and let $\chi_d(.)$ be the real quadratic character associated with $d$. Thus, $\chi_d$ is the Kronecker symbol $\chi_d(n)=\left(\frac d n \right )$. Alternatively, $\chi_d$ is a real, primitive character with modulus $\vert d\vert$; this completely specifies $\chi_d$ except when $8\mid d$, in which case there are either 0 or 2 such characters. If we note that $\chi_d$ is an even character when $d>0$ and is odd when $d<0$ then the ambiguity is removed. Finally, we note that the Dedekind zeta-function associated with the field $K=Q(\sqrt{d})$ is given by

$$\zeta_K(s)=\zeta(s)L(s,\chi_d).$$

We are interested in the order of vanishing of the quadratic twist

$$L_f(s,\chi_d)=\sum_{n=1}^\infty \frac{a_n \chi_d(n)}{n^s}$$

at the center $s=1/2.$

If $f$ is a modular form of weight 2 associated with an elliptic curve $E/Q$ given by $E:y^2=x^3+A x+B$ then the $L$-function of $E$ is $L_E(s)=L_f(s)$ and the twisted $L$-function $L_E(s,\chi_d)$ is just the $L$-function of another elliptic curve, called the twisted elliptic curve $E_d:dy^2=x^3+A x+B$. Then our question about the order of the zero of $L_E(s,\chi_d)$ is, under the asumption of the Birch and Swinnerton-Dyer conjecture, the same as the question of the distribution of ranks in the family $E_d$ of twisted elliptic curves

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