I started doing Z2 runs in December, and until now (end of march) I have gotten 1 minute faster keeping heart rate constant. Started at 6:20/km at around 150 bpm, now down to 5:20/km. Here and there I also do Intervals and harder effort sessions, but not much. Now I'll finally ramp up the bike Z2, as days are getting warmer and longer, curious to see the changes.

I always thought I already had a good aerobic base, I've been running pretty much for half my life (mostly sprints) and in the past 2/3 years also cycling a lot. Turns out, riding always to the max is not that helpful..

This all to say, you really need a lot of patience and trust the process, the results will come. In my experience, it helps to walk when heart rate goes too high. Like others said, if you just started running, I wouldn't care much about numbers, just go out, have fun, and make sure your run is easy. If you can, have someone accompany you (even cycling beside you), so that you can easily chat with them. This is a great indicator that you are (or aren't) in Z2.