code: def z_offline(ctx, w, k=0):
      r"""
      Computes Z(w) and its derivatives off the line
      """
      s = ctx.mpf("0.5") + ctx.j * w
      s1 = s
      s2 = ctx.conj(1 - s1)
      wpinitial = ctx.prec
      ctx.prec = 35
      #  X see II Section 3.21 (109) and (110)
      # M1  see II Section 3.21 (111) and (112)
      if ctx._re(s1) >= 0:
          M1 = 2 * ctx.sqrt(ctx._im(s1) / (2 * ctx.pi))
          X = ctx.sqrt(abs(s1))
      else:
          X = (2 * ctx.pi) ** (ctx._re(s1) - 1) * abs(1 - s1) ** (
              0.5 - ctx._re(s1)
          )
          M1 = 4 * ctx._im(s1) * X
      # M2  see II Section 3.21 (111) and (112)
      if ctx._re(s2) >= 0:
          M2 = 2 * ctx.sqrt(ctx._im(s2) / (2 * ctx.pi))
      else:
          M2 = (
              4
              * ctx._im(s2)
              * (2 * ctx.pi) ** (ctx._re(s2) - 1)
              * abs(1 - s2) ** (0.5 - ctx._re(s2))
          )
      # T  see II Section 3.21  Prop. 27
      T = 2 * abs(ctx.siegeltheta(w))
      # defining some precisions
      # see II Section 3.22 (115), (116), (117)
      aux1 = ctx.sqrt(X)
      aux2 = aux1 * (M1 + M2)
      aux3 = 3 + wpinitial
      wpbasic = max(
          6, 3 + ctx.mag(T), ctx.mag(aux2 * (26 + 2 * T)) + aux3
      )
      wptheta = max(4, ctx.mag(2.04 * aux2) + aux3)
      wpR = ctx.mag(4 * aux1) + aux3
      # now the computations
      ctx.prec = wptheta
      theta = ctx.siegeltheta(w)
      ctx.prec = wpR
      xrz, yrz = Rzeta_simul(ctx, s, k)
      pta = 0.25 + 0.5j * w
      ptb = 0.25 - 0.5j * w
      if k > 0:
          ps1 = (
              0.25 * (ctx.psi(0, pta) + ctx.psi(0, ptb))
              - ctx.ln(ctx.pi) / 2
          )
      if k > 1:
          ps2 = (1j / 8) * (ctx.psi(1, pta) - ctx.psi(1, ptb))
      if k > 2:
          ps3 = (-1.0 / 16) * (ctx.psi(2, pta) + ctx.psi(2, ptb))
      if k > 3:
          ps4 = (-1j / 32) * (ctx.psi(3, pta) - ctx.psi(3, ptb))
      ctx.prec = wpbasic
      exptheta = ctx.expj(theta)
      if k == 0:
          zv = exptheta * xrz[0] + yrz[0] / exptheta
      j = ctx.j
      if k == 1:
          zv = (
              j * exptheta * (xrz[1] + xrz[0] * ps1)
              - j * (yrz[1] + yrz[0] * ps1) / exptheta
          )
      if k == 2:
          zv = exptheta * (
              -2 * xrz[1] * ps1
              - xrz[0] * ps1**2
              - xrz[2]
              + j * xrz[0] * ps2
          )
          zv = (
              zv
              + (
                  -2 * yrz[1] * ps1
                  - yrz[0] * ps1**2
                  - yrz[2]
                  - j * yrz[0] * ps2
              )
              / exptheta
          )
      if k == 3:
          zv1 = (
              -3 * xrz[1] * ps1**2
              - xrz[0] * ps1**3
              - 3 * xrz[2] * ps1
              + j * 3 * xrz[1] * ps2
          )
          zv1 = (
              (zv1 + 3j * xrz[0] * ps1 * ps2 - xrz[3] + xrz[0] * ps3)
              * j
              * exptheta
          )
          zv2 = (
              3 * yrz[1] * ps1**2
              + yrz[0] * ps1**3
              + 3 * yrz[2] * ps1
              + j * 3 * yrz[1] * ps2
          )
          zv2 = (
              j
              * (zv2 + 3j * yrz[0] * ps1 * ps2 + yrz[3] - yrz[0] * ps3)
              / exptheta
          )
          zv = zv1 + zv2
      if k == 4:
          zv1 = (
              4 * xrz[1] * ps1**3
              + xrz[0] * ps1**4
              + 6 * xrz[2] * ps1**2
          )
          zv1 = (
              zv1
              - 12j * xrz[1] * ps1 * ps2
              - 6j * xrz[0] * ps1**2 * ps2
              - 6j * xrz[2] * ps2
          )
          zv1 = (
              zv1
              - 3 * xrz[0] * ps2 * ps2
              + 4 * xrz[3] * ps1
              - 4 * xrz[1] * ps3
              - 4 * xrz[0] * ps1 * ps3
          )
          zv1 = zv1 + xrz[4] + j * xrz[0] * ps4
          zv2 = (
              4 * yrz[1] * ps1**3
              + yrz[0] * ps1**4
              + 6 * yrz[2] * ps1**2
          )
          zv2 = (
              zv2
              + 12j * yrz[1] * ps1 * ps2
              + 6j * yrz[0] * ps1**2 * ps2
              + 6j * yrz[2] * ps2
          )
          zv2 = (
              zv2
              - 3 * yrz[0] * ps2 * ps2
              + 4 * yrz[3] * ps1
              - 4 * yrz[1] * ps3
              - 4 * yrz[0] * ps1 * ps3
          )
          zv2 = zv2 + yrz[4] - j * yrz[0] * ps4
          zv = exptheta * zv1 + zv2 / exptheta
      ctx.prec = wpinitial
      return zv
  
file_path: catboost/contrib/python/mpmath/py3/mpmath/functions/rszeta.py
imports: import math
  from .functions import defun
label_desc: Use image node_id 10 for calling a global function with example usage: z_offline(ctx, w, k) and returns: zv