Let $\Phi_1,\Phi_2$ be Maass forms for $\text{ SL}(2,\mathbb Z)$ with Fourier coefficients $C_1(n),C_2(n)$. For a positive integer $h$ the meromorphic continuation and growth in $s\in\mathbb C$ (away from poles) of the shifted convolution L-function $$L_h(s,{ \Phi_1,\Phi_2})\, := \sum_{ n \neq 0,-h} { C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{ -\frac{ 1}{ 2}s}$$ is obtained. For ${ \rm Re}(s) > 0$ it is shown that the only poles are possible simple poles at $\frac{ 1}{ 2} \pm ir_k$, where $\tfrac14+r_k^2$ are eigenvalues of the Laplacian. As an application we obtain, for $T\to\infty$, the asymptotic formula \begin{ align*} & \underset{ n \neq 0,-h}{ \sum_{ \sqrt{ |n (n + h)|}<T} } \hskip-5pt{ C_1(n) C_2(n + h)} \left(\text{ log}\Big(\tfrac{ T}{ \sqrt{ |n (n + h)|}}\,\Big)\right)^{ \frac{ 3}{ 2} + \varepsilon} \hskip-7pt =\; f_{ { \mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \cdot T^{ \frac{ 1}{ 2}} \; + \; \mathcal O\left( h^{ 1-\varepsilon} T^\varepsilon + h^{ 1 + \varepsilon} T^{ -2 - 2\varepsilon} \right), \end{ align*} where the function $f_{ { \mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T)$ is given as an explicit spectral sum that satisfies the bound $f_{ { \mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \ll h^{ \theta + \varepsilon}$. We also obtain a sharp bound for the above shifted convolution sum with sharp cutoff, i.e., without the smoothing weight $\log(*)^{ \frac32+\varepsilon}$ with uniformity in the $h$ aspect. Specifically, we show that for $h < x^{ \frac{ 1}{ 2} - \varepsilon}$, \[ { \sum_{ \sqrt{ |n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{ \frac{ 2}{ 3}\theta + \varepsilon}x^{ \frac{ 2}{ 3} (1 + \theta) + \varepsilon} + h^{ \frac{ 1}{ 2} + \varepsilon}x^{ \frac{ 1}{ 2} + 2\theta + \varepsilon}. \]
Categories: math.NT