A complex number and its conjugate have the same absolute value, then (x+yi)*(x-yi) = (sqrt(441000))^2 which simplifies to x^2+y^2 = 441000.

Then our problem can be asked to determine the number of Gaussian conjugates pairs that 441000 can be factored into (where order matters).

Gaussian integers have unique factorization, similar to regular integers.  But there are now four multiplicative units 1, i, -1, -i.  And we'll require our Gaussian primes to have its real part be an odd integer.

We can start with the regular prime factorization of 441000 = 2^3 * 3^2 * 5^3 * 7^2.

A potential Gaussian factor can be  multiplied by one of the four units 1, i, -1, -i.  

Four possible choices for x+yi

2 is composite over Gaussian integers, specifically 2 = -i * (1+i)^2.  So 2^3 contributes six factors of (1+i).  For the conjugate factorizations we seek, (1+i)^3 will allocated to each factor.  

One possible choice for x+yi

All regular primes of the form 4k+3 are still prime in Gaussian integers.  So 3^2 and 7^2 must attribute exactly one 3 and one 7 to each conjugate factor.  Note that if either 3 or 7 was raised to an odd power then the conjugate factorization sought would be impossible.

One possible choice for x+yi

Finally, primes of the form 4k+1 are all composite products of a pair of conjugate Gaussian primes.  5 = (1+2i) * (1-2i); then 5^3 = (1+2i)^3 * (1-2i)^3.  In this case there are four (one more than the exponent 3) distinct ways to split the six available factors between the conjugates: (1+2i)^3, (1+2i)^2 * (1-2i), (1+2i) * (1-2i)^2, (1-2i)^3

Four possible choices for x+yi

So total choices = 4*1*1*4 = 16.  The 16 values can be parameterized as 21* (1+i)^3 * i^m * (1+2i)^n * (1-2i)^(3-n) for m in {0,1,2,3} and n in {0,1,2,3}.

An example m=1 and n=1: 21 * (1+i)^3  * i * (1+2i) * (1-2i)^2 = -630 + 210i.  This corresponds to the point (-630, 210) in the original problem statement.

Then backtracking to the original problem statement: A circle with radius sqrt(441000) centered at the origin passes through 16 lattice points.