  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.
