Imagine a species achieving immortality. There will probably a transition period with a mortal and immortal population first. But the immortals should quickly dominate the mortals due to their immortality advantage.
What would be their primary motivation? I think it would be self preservation first and foremost. Self preservation also means resource conservation. There would be no point in having offsprings as they would compete with you for resources. You also would want to get rid of the mortal resource wasters.
Now imagine a first species, becoming immortal, stopping growth, being mostly concerned with their own existence for millions or even billions of years. It makes me think of Lovecraft’s Great Old Ones. And this is where things get a bit disturbing.
How would they perceive any new life arising in the universe? There might of course be some curiosity at first, especially if the species is different from them. But they’d probably want to get rid of it at some point, seeing it as a rival for resources.
So maybe that is the reason we are not getting any signals from aliens. And maybe we are just lucky to not have piqued their interest yet…