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Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g, s)$ be an Eisenstein series on $GL_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $GL_2(\mathbb{F})$. We will consider in this paper the following automorphic integral: $$\int_{Z_{A}GL_{2}(\mathbb{F})\backslash GL_{2}(\mathbb{A}_{\mathbb{F}})} F(g)E(g,s) dg.$$ This is in some sense the complementary case to the well-known Rankin-Selberg integral and the triple product formula. We will approach this integral by Waldspurger's formula. We will discuss when the integral is automatically zero, and otherwise the L-function it represents. We will calculate local integrals at some ramified places, where the level of the ramification can be arbitrarily large.

Cuspidal part of an Eisenstein series restricted to an index 2 subfield