It took a little over a week. A few days after I put in an application, an HR rep gave me a call to gauge if I was a good enough match to send over to the hiring manager. Once that was out of the way, I was scheduled for a phone interview with the hiring manager a few days later.
The questions focused mostly on technical skill set and experience and took about 45 minutes, though we'd originally only scheduled 30. I could tell he was very knowledgeable and we were able to have a comfortable discussion on the position and what it would entail. At the end of the interview, I was told he would have the HR rep schedule a behavioral based interview with me as the next step and gave me some background details on it.
I never heard back after this point. I'm a little surprised they didn't at least give a rejection email based on the previous email behavior with HR.