Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = aX^2+bXY+cY^2$ of discriminant $b^2-4ac = -D$ (thus $\mathrm{gcd}(a,b,c) = 1$, $-a < b \leq a \leq c$). Those are the ordinates of the CM points $z_Q = \frac{b + \sqrt{-D}}{2a}$ in the standard fundamental domain for $\mathcal{H} / \mathrm{PSL}(2,\mathbb{Z})$. By Duke's theorem, the $z_Q$ are equidistributed in the measure $\frac{3}{\pi} \frac{dx \, dy}{y^2}$. The highest lying point has ordinate $k := \sqrt{|D|}/2$, and since $\int_1^k t \frac{dt}{t^2} = \log{k}$, equidistribution suggests (!) an expectation of $\log{\sqrt{|D|}} = \log{k} + O(1)$ for the mean value of $\frac{\pi}{3} y_Q$.

This heuristic cannot be turned into a proof since the equidistribution result only applies to a compactly supported test function (and our mean value is much larger than $\log{|D|}$ under a putative 'Siegel zero'); indeed, it is probably not a priori clear if such an expectation is reasonable. Nonetheless, this can be proved by virtue of the limit formula of Kronecker, Chowla and Selberg, which gives $\mathrm{Avg}_Q (\frac{\pi}{3} y_Q - \log{y_Q} )= \log{\sqrt{|D|}} + \frac{L'}{L}(1,\chi_D) + O(1)$ (and the mean value of $\log{y_Q}$ can be proved to be bounded under ERH). Hence our expected $\sim \log{k}$ asymptotic follows under ERH, and we do know that the heuristic is right in this particular case.

This direct (Kronecker) relation of the mean value of $\varphi(\frac{\pi}{3}y_Q)$ to the zeros of $L(s,\chi_D)$ is unique to $\varphi(\frac{\pi}{3}y) = -4\log{|\eta(q)|}$, a constant multiple of the logarithm of the Dedekind eta function, which is approximately a linear function in $y$ (asymptotically this function is $\frac{\pi}{3} y - \log{y} +O(e^{-y})$ as $y \to \infty$). This is what allows to treat the linear case $\varphi(y) = y$, in addition to the compactly supported functions $\varphi$. But (presumably?) the same heuristic would suggest $\int_1^k \varphi(t) \frac{dt}{t^2}$ for the expected value, for certain other reasonable unbounded functions $\varphi$.

Do we have $\mathrm{Avg}_Q \frac{\pi}{3} y_Q \log{y_Q} \sim \frac{1}{2} (\log{k})^2$ and $\mathrm{Avg}_Q (\frac{\pi}{3} y_Q)^2 \sim k$, assuming GRH and possibly other mainstream analytic hypotheses? Are there results out there that will allow to compute a GRH-conditional asymptotic for these averages?

Of course, the same question could be asked for the positive discriminants (indefinite binary quadratic forms and closed geodesics on the modular surface).