Continuous transformation of a q-expansion into a Dirichlet series

Question

I would like to know if series of the form $ \sum_{n>0}a_{n}\dfrac{q^{tn}}{n^{z(1-t)}} $ where $ t\in(0,1) $ , $ a_{n} $ is the $ n $ -th Fourier coefficient of a cusp form of a given weight $k $ and level $ N $ and $q=e^{2i\pi z} $ for $ z $ in the upper half plane of complex numbers with positive imaginary part have been considered so far. If so, is the value of (the derivative of) such a series for $ t=1/2 $ of interest ? Can an analogue of the Rankin-Selberg convolution be defined for such series ?