nt.number theory - Positivity of $L(1,chi)$ for real Dirichlet's character

Let $chi$ be a real nonprincipal Dirichlet's character modulo $m$.

In my
answer to the question on $L(1,chi)$, I explain a trick for showing that $L(1,chi)>0$ on the simplest examples
of the real characters modulo 3 and 4. The proof goes as follows: one takes
$$
f(x)=sum_{n=1}^inftychi(n)x^n=frac1{1-x^m}sum_{j=1}^{m-1}chi(j)x^j
$$
and uses Abel's theorem to write
$$
L(1,chi)=int_0^1f(x)dx;
$$
since the corresponding function $f(x)$ is positive on $(0,1)$, the latter integral has to be positive.

Clearly, $1-x^m>0$ on $(0,1)$, so that the required positivity of $f(x)$ reduces to the positivity of
the polynomial
$$
g_chi(x)=sum_{n=1}^{m-1}chi(n)x^n
$$
on $(0,1)$. Trying to verify on how generalizable is this method for $m>3$, I was quite surprised to see that it works
perfectly further; for example,
$$
g(x)=x(1-x)(1-x^2)>0 quadtext{if } m=5
$$
or
$$
g(x)=x(1-x)(1+x^2+2x^3+3x^4+2x^5+x^6+x^8)>0 quadtext{if } m=11.
$$
Honestly saying, the positivity is not so obvious in many other examples (for example, $m=19$) but nevertheless it is
always holds for small values $mle30$.

Question. Given an integer $m>2$ and a real nonprincipal character $chi$ modulo $m$,
is it true that $g_chi(x)>0$ for $xin(0,1)$? If not, are there (in)finitely many $m$ for
which the positivity does not take place? Is the above strategy for showing $L(1,chi)ne0$ discussed
in the literature?