I would like to share a new app with you. It´s called Towelroot. It roots your Device instantly.
Just go to http://towelroot.com and click on the lambda. Install it and click make it rain.
It Works!! No need for Recovery or anything else. Locked Bootloader AND Unlocked Bootloader.
My Z1 runs the latest 4.4.2 so it works with the latest firmware.
Geohot (creator of towel root) says that it roots nearly every device. So keep on sharing.
P.S: You have to install SuperSU/Superuser or whatever app you like manually.