For a normal diode, the thin line on the component corresponds with the line of the symbol. Is this the same for a zener diode?
Yes, it's the cathode. Don't take my word for it, it's easily tested with a multimeter; the voltage drop with this line negative should be 0.6V or so, the same as any other Si diode. To see the zener voltage, connect it reverse biassed (thin line positive)