------
## 最终结论
```python
def odd_Equivalent(binary_string, length):
    count = 0
    for i in range(length):
        rotated_string = binary_string[i:] + binary_string[:i]
        if int(rotated_string, 2) % 2 == 1:
            count += 1
    return count

# Test cases
assert odd_Equivalent("011001", 6) == 3
assert odd_Equivalent("11011", 5) == 4
assert odd_Equivalent("1010", 4) == 2
```