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Could you suggest me a reference where the following non-holomorphic generalization of the Eisenstein series is discussed?

$$ G_{k,l}(\tau,z) = \sum_{m,n} (z+m+n\tau)^{-k}(\bar z+m+n\bar \tau)^{-l} $$

When $l=0$ and $z=0$ (and removing the sum on $(m,n)=(0,0)$) it's the standard Eisenstein series.

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You can also express this in terms of the series $\sum_{m, n} (z + m + n\tau)^{-k} (\Im \tau)^s |z + m + n\tau|^{-2s}$, which is a very well-studied family of non-holomorphic weight $k$ Eisenstein series, closely connected with L-functions of modular forms. E.g. there is a chapter on these in Miyake's book, and also in Hida's blue book "Elementary theory of L-functions and Eisenstein series". – David Loeffler Dec 17 '12 at 11:40

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