A somewhat amusing example in the theory of automorphic forms is that the constant $1$ function is the residue of an Eisenstein series (considered as a meromorphic vector-valued function). So, any $1$, anywhere, can be replaced by $Res_{s=1}E_s$.

As a quick application, take a cusp form $f$ for $GL_2$ over a number field $k$ that generates an irreducible representation $\pi_f$. Identifying $1$ with the residue of an Eisenstein series, the $L^2$-norm of $f$ turns into the residue of a Rankin-Selberg L-function times the norm of the first Fourier-Whittaker coefficient of $f$, $\rho_f(1)$: $$||f||_{L^2}=\int_X |f|^2\ dx=Res_{s=1}\int_X |f|^2E_s\ dx=|\rho_f(1)|^2Res_{s=1}\Lambda(s,\pi_f\otimes\tilde \pi_f)$$ If we want, we can break up the Rankin-Selberg L-function a bit to get something more tangible. $$||f||_{L^2}=|\rho_f(1)|^2L_\infty(1,\pi_f\otimes\tilde \pi_f) L(1,Sym^2\pi_f)Res_{s=1}\zeta_k(s)$$ where $L_\infty$ is a certain product of Gamma functions (whose parameters depend on $\pi_f$), $\zeta_k$ is the Dedekind zeta function of $k$ (whose residue at $s=1$ we know from the class number formula), and $L(1,Sym^2\pi_f)$ is the symmetric-square L-function of $\pi_f$, which is more mysterious (though a certain amount is known about how it changes as $\pi_f$ varies). Typically, we assume either $||f||_{L^2}=1$ or $\rho_f(1)=1$, so the formula turns information about L-functions into information about Fourier-Whittaker coefficients or $L^2$-norms (or vice-versa).