Average first negative value conjecture for Rankin–Selberg Fourier coefficients

Let $D_1$ and $D_2$ range over fundamental discriminants with $|D_1D_2|\leq x$, and let $p_1<p_2<\cdots$ be the primes. Define $\eta(D_1,D_2)$ to be the smallest $p_i$ such that

$$\operatorname{sgn}(\sigma_{\chi_1,\chi_2}^{k-1}(p_i))=-1.$$

Average first negative value conjecture. As $x\to\infty$, one expects

$$\frac{\sum_{|D_1D_2|\leq x}\eta(D_1,D_2)}{\sum_{|D_1D_2|\leq x}1}\longrightarrow\theta,$$

where

$$\theta:=\sum_{\ell=1}^{\infty}\frac{p_\ell^2(p_\ell+2)}{2(p_\ell+1)^2}\prod_{i=1}^{\ell-1}\frac{2+p_i(p_i+2)}{2(p_i+1)^2}.$$

The conjecture predicts that the average first prime at which the relevant Fourier coefficient has negative sign converges to the numerical constant $\theta=3.9750223902667539847734759105175510246019355513991\ldots$.