[exim] Is the $primary_hostname automatically local?

Hi guys,

I run exim within the DirectAdmin control panel environment.
This is exim 4.24

Looking at the config file I see the following:

========================================
domainlist local_domains = lsearch;/etc/virtual/domains
                

domainlist relay_domains = lsearch;/etc/virtual/domains : \
                localhost

hostlist relay_hosts = net-lsearch;/etc/virtual/pophosts : 127.0.0.1
hostlist auth_relay_hosts = *

primary_hostname = my.host.machine

========================================

Now I was just wondering if I need to put " : my.host.machine " at the
end of the local_domains list.

Or is the primary_hostname always appended internally to the
local_domains list?

Thanks for your help,
Daniel