Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) :

$$P(\chi,N)=\prod_{i=1}^{N} \frac{1}{1-\chi(p_i) p_i^{-s}} $$

According to my knowledge, this product (for $0<Re(s)<1$) is not known to converge. I expect it to have a quite erratic behavior (for reason explained here - concerning Zeta case : Is the Euler product formula always divergent for 0<Re(s)<1?)

But can we expect $|P(\chi,N)|$ to be lower than a fixed constant for an infinity of N values ? What is known on the behavior of this partial product ?