# Imaginary Zeros of Zeta Function

I was doing some work with the zeta function and have a question.

I am aware that the Riemann Hypothesis claims that all of the critical zeros of the analytically continued zeta function have a real part Re(z)=1/2.

My question is, does the concept apply only to the complex zeros, or the imaginary and real parts separately.

Basically, is it possible to have:

Im(zeta(z))=0

Without having:

Re(zeta(z))=0

Or does a zero of one part automatically illustrate the existence of a zero for the other?

A zero x of a function f is when f(x)=0, (=0+0i) and it is no different with the zeta function.

Or does a zero of one part automatically illustrate the existence of a zero for the other?

No. For example, along the real line the imaginary part of the zeta function is zero, but the real part is certainly not always zero.