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I was trying to compare the basicity of o-phenanthroline and ammonia.

According to me, the factors that are seen in o-phenanthroline are:

  1. o-Phenanthroline has two nitrogen atoms at the 1,10 positions.
  2. It has a free lone pair on one nitrogen that is not delocalized due to resonance.
  3. The nitrogens are $\mathrm{sp^2}$ hybridized which is more electronegative compared to the $\mathrm{sp^3}$ lone pair on ammonia.

However, would there be hydrogen bonding in its conjugate base? If so, then how do we compare its basicity with a base like ammonia?

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A quote from here explains and give a good answer to your wondering:

Amines are the most basic of the common organic functional groups, but are still fairly weak bases. Protonation occurs on the non-bonded electron pair exclusively. The basicity of amines is directly dependent on the “electron density” at the nitrogen atom. Both inductive and resonance effects can alter the basicity of a nitrogen atom.

Hybridization on the $\ce{N}$ also affects basicity. An increase in $\mathrm{s}$ character on an atom increases the electronegativity of that atom which favors acidity and therefore disfavors basicity. Hence $\mathrm{sp^3}$-hybridized nitrogen is more basic than either $\mathrm{sp^2}$ or $\mathrm{sp}$ hybridized nitrogen.

The availability of this non-bonding lone pair is a factor of basisity. The 1,10-Phenanthroline is a pyridine derivative. Thus, both lone pairs of two $\ce{N}$ atoms in the ring system are contributed to the system's aromaticity. This make them not 100% available to incoming protons while the lone pair in ammonia is 100% available. Therefore, in aqueous solutions, the basicity of $\ce{NH3} \ (\mathrm{p}K_\mathrm{a} = 9.3)\gt$ basicity of 1,10-Phenanthroline $(\mathrm{p}K_\mathrm{a} \approx 4.9)$. This is similar to $\mathrm{p}K_\mathrm{a}$ of pyridine, which is $5.2$ in water (comparison with piperidine is depicted in following scheme):

Pyridine and piperidine

Keep in mind that there are some other factors are effecting the basicity of amine as well. For example, the ring sizes of cyclic amines (Ref.1):

pKa of cyclic amines

The order is 5-membered $\ge$ 4-membered $\gt$ 6-membered $\gg$ 3-membered.

References:

  1. Scott Searles, Milton Tamres, Frank Block, Lloyd A. Quarterman, "Hydrogen Bonding and Basicity of Cyclic Imines," J. Am. Chem. Soc. 1956, 78(19), 4917–4920 (https://doi.org/10.1021/ja01600a029).
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