The holomorphy and shadow conjecture for weight $1/2$ Eisenstein series

Let $(\Lambda'/\Lambda,q)$ be a discriminant form, and let $E_{1/2}^*(\tau,s)$ denote the weight $1/2$ Eisenstein series associated with it. For the discriminant form $(\Lambda'/\Lambda,-q)$, let $E_{3/2,\beta}(\tau)$ denote the mock Eisenstein series indexed by $\beta$.

Holomorphy and shadow conjecture. The value $E_{1/2}^*(\tau,0)$ is a holomorphic modular form of weight $1/2$; moreover, it is a linear combination of the shadows of the mock Eisenstein series $E_{3/2,\beta}(\tau)$ for $(\Lambda'/\Lambda,-q)$ whose constant term is $1\cdot\mathfrak{e}_0$.

This conjecture would provide the expected analytic continuation and modular interpretation of the weight $1/2$ Eisenstein series, where the direct Fourier-expansion approach encounters a singularity. It is motivated by the relation with the Bruinier–Funke operator applied to weight $3/2$ Eisenstein series and by the anticipated functional equation, but the statement is presented as conjectural here.