How can I plot this Hurwitz Zeta-based function at negative arguments?

Question

Here is the code:

pw[y_, x_] := -HurwitzZeta[1 - x, 1/2 + y] x
Plot3D[pw[b, p], {b, -3, 3}, {p, -3, 3}, Mesh -> {6, 5}, 
 ClippingStyle -> None, 
 PlotStyle -> Directive[Orange, Opacity[.8], Specularity[White, 20]], 
 PlotPoints -> 100, AxesLabel -> Automatic]

Here is what I get:

How can I plot this function at $b<-1$? Numerically it evaluates well.

Answer 1

Clear@pw

pw[x_, y_] := -HurwitzZeta[1 - y, 1/2 + x] y

With negative x-values you get many complex numbers and indeterminate values. For example:

pw[-3., -2.5] // N

29.052 + 28.9903 I

pw[-3., 0.] // N

Indeterminate

Avoiding negative x-values gives the following plot

Plot3D[pw[x, y], {x, 0, 3}, {y, -3, 3}]

If you want to plot negative x-values you must skip the 0 by choosing a plot range of f.e. {-3.01, 3.01} and you can convert the complex numbers with Abs

Plot3D[Abs @ pw[b, p], {b, -3.01, 3.01}, {p, -3.01, 3.01}]

From the documentation:

Unlike Zeta, HurwitzZeta has singularities at a==-n for non-negative integers n ... HurwitzZeta has branch cut discontinuities in the complex a plane running from 0 to -Infinity.