Classically, we can attach $L$-functions (with properties like, analytic continuation, functional equation) to Dirichlet characters, Hecke eigenforms, etc... 


My question is: can one attach $L$-functions (properties similar to the above) to Half-Integral weight modular eigenforms as well? If they do exist, can someone
provide a reference for this?



Do L-functions exist for Half-integral weight modular forms?
Since the original question asked only for analytic continuation, functional equation, I'd like to add that the Mellin transform $\sum a\_n n^{-s}$ of the half integral weight modular form $\sum a\_n \exp(2 \pi i n z)$ has
these two properties, but it lacks an Euler product decomposition even if the modular form is a Hecke eigenform, since the Fourier coefficients at square free indices are not multiplicatively related.
A similar phenomenon occurs for Siegel modular forms with the so called Koecher-Maaß series; here the Fourier coefficients at matrices representing maximal lattices (Kern- resp. Stammformen in Brandt's terminology) have no multiplicative relation. 


