Linear viscoelastic and dielectric behavior was examined for a diblock copolymer composed of cis-polyisoprene (PI) and poly(p-tert-butylstyrene) (PtBS) blocks of the molecular weights M-PI = 52.(6) x 10(3) and M-PtBS = 41.(8) x 10(3) (PI weight fraction W-PI = 55.7 wt %). These blocks were miscible and the copolymer was in the disordered state at temperatures examined, 20 <= 5 T/degrees C <= 120. PI has the type-A dipole parallel along the chain backbone while FtBS does not. Thus, the dielectric response at low frequencies exclusively detected the global motion (end-to-end vector fluctuation) of the PI block, while the viscoelastic response reflected the motion of the copolymer chain as a whole. The dielectric data of the PI block exhibited prominent thermo-theological complexity. Since the PI and PtBS blocks behaved as the fast and slow blocks at low T and this difference of their relaxation rates decreased at high T, the complexity was related partly to the dynamic frictional heterogeneity for the PI block resulting from this dynamic asymmetry of the two blocks. However, it turned out that the complexity was more importantly related to the connectivity between the PI and PtBS blocks: Namely, the PI block possibly behaved as a tethered chains at low T (where the slow PtBS block effectively anchored the PI block) and as a portion of a five linear chain at high T (where this anchoring effect vanished), and this change in the motional mode of the PI block appeared to dominate the thermo-rheological complexity of the PI block. For a test of this molecular picture, a PI/PtBS blend having the same w(Pt) as the copolymer was utilized as a reference to reduce the copolymer data at an iso-relaxation-time (iso-T-s) state defined for the Rouse segment of PI in the copolymer and bulk systems. It turned out that the dielectric data of the copolymer at low T were dose to literature data for star ranched bulk PI, while the copolymer data at high T were dose to the data expected for a linear bulk PI chain having the type-A dipole only in a portion of its backbone and feeling an extra friction due to the other portion (corresponding to PtBS). These results lent support to the above molecular picture. The slow dynamics of the copolymer was dominated by the PtBS block and thus exhibited thermo-rheological complexity weaker than that of the PI block.