  In this work we study the Plancherel-Rotach type asymptotics for $q$-Laguerre
orthogonal polynomials with complex scaling . The main term of the asymptotics
contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the
vertical line $\Re(s)=2$, while the main term of the asymptotics involves the
theta function for the scaling parameter in the strip $0<\Re(s)<2$. In the
latter case the number theoretical property of the scaling parameter completely
determines the order of the error term. $ $These asymptotic formulas may
provide insights to some new random matrix model and add a new link between
special functions and number theory.
