Question: What battery reference is suitable to fit in my DCH-Z510?

Answer: Your DCH-Z510 supports BL-5C batteries.

    Type: Li-Ion
    Capacity: 970 - 1050 mAh
    3.7 V
    3.885 Wh (1050mAh model)
    GND - BSI resistance: 82 kOhm
    Size: 53 mm x 34 mm x 6 mm
    Weight: 23 g

You can easily find them on Internet stores by looking for the reference BL-5C.