Let $h$ be a class number of an imaginary quadratic number field of discriminant $d$. It holds that $h = k(d)\cdot L_d(1)$ where $k(d)$ is the Dirichlet structure constant and $L_d$ is the Dirichlet $L$-function. I have recently heard that $L_d(1)$ can be bounded by $\log(d)$ but I can't find any source for this since I have no background in analytic number theory. Can anyone direct me to a book/paper/anything which I can use as reference?
Follow-up question: Is there a better bound for the class number which is efficiently computable?
For a non-trivial primitive Dirichlet character $\bmod d$ we have the Polya-Vinogradov inequality $$\sum_{n\le x} \chi(n) = O(d^{1/2}\log d)$$ (the $O$-constant doesn't depend on $x,\chi,d$)
Then $$L(1,\chi) =\sum_{n\ge 1} \chi(n) n^{-1}=\sum_{n\ge 1} \chi(n) \int_n^\infty x^{-2}dx= \int_1^\infty (\sum_{n\le x}\chi(n))x^{-2}dx$$ $$= \int_1^{d^{1/2}} O(x)x^{-2}dx+\int_{d^{1/2}}^\infty O(d^{1/2}\log d) x^{-2}dx=O(\log d)+O(\log d)$$