Selberg's orthonormality conjecture for the Selberg class

Let ${\mathcal S}$ be the Selberg class of functions $L(s)=\sum_{n=1}^{\infty}a_L(n)n^{-s}$ satisfying the Ramanujan hypothesis, analytic continuation, a functional equation, and an Euler product. A function in ${\mathcal S}$ is primitive if it cannot be written as a nontrivial product of functions in ${\mathcal S}$. For $L\in{\mathcal S}$, write $a_L(p)$ for its Dirichlet-series coefficient at a prime $p$. Selberg's orthonormality conjecture. For every function $1\ne L\in{\mathcal S}$, there is a positive integer $\kappa_L$ such that

$$\sum_{p\leq x}\frac{|a_L(p)|^2}{p}=\kappa_L\log\log x+R(x),$$

and, for any primitive functions $L_1,L_2\in{\mathcal S}$,

$$\sum_{p\leq x}\frac{a_{L_1}(p)\overline{a_{L_2}(p)}}{p}=R(x),$$

where $R(x)\ll 1$. The first relation predicts the logarithmic mean square of prime coefficients, while the second is the orthonormality relation for distinct primitive $L$-functions. This conjecture is closely related to the prime-coefficient condition used in universality results for $L$-functions, but its general validity for the Selberg class remains open.