  For an irreducible orientable compact $3$-manifold $N$ with empty or
incompressible toral boundary, the full $L^2$--Alexander torsion
$\tau^{(2)}(N,\phi)(t)$ associated to any real first cohomology class $\phi$ of
$N$ is represented by a function of a positive real variable $t$. The paper
shows that $\tau^{(2)}(N,\phi)$ is continuous, everywhere positive, and
asymptotically monomial in both ends. Moreover, the degree of
$\tau^{(2)}(N,\phi)$ equals the Thurston norm of $\phi$. The result confirms a
conjecture of J.~Dubois, S.~Friedl, and W.~L\"uck and addresses a question of
W.~Li and W.~Zhang. Associated to any admissible homomorphism
$\gamma:\pi_1(N)\to G$, the $L^2$--Alexander torsion
$\tau^{(2)}(N,\gamma,\phi)$ is shown to be continuous and everywhere positive
provided that $G$ is residually finite and $(N,\gamma)$ is weakly acyclic. In
this case, a generalized degree can be assigned to $\tau^{(2)}(N,\gamma,\phi)$.
Moreover, the generalized degree is bounded by the Thurston norm of $\phi$.
