TITLE: Critical points of Dirichlet L functions
QUESTION [11 upvotes]: Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal
character $\chi$.  This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached through real values of $s$, $1 < s < \infty$ . 
For example, let $\chi_4$ be the non-trivial character $(\mathrm{mod}\ 4)$ taking values $(1,0,-1,0)$ at $(1,2,3,4)$. We have $L(1,\chi_4) < 1$ and, by an alternating series argument, $L(s,\chi_4)$ is monotonically increasing for $s > 1$.
Next consider $\chi_7$ $(\mathrm{mod}\ 7)$ taking values $(1,1,-1,1,-1,-1,0)$ at $(1,2,3,4,5,6,7)$. Note that $L(1,\chi_7) > 1$. However, $L(s,\chi_7)$ is not monotone decreasing for $s > 1$, but first increases with a peak near $s = 1.1$ before decreasing. What can one say in general?
Questions: 1) Are there infinitely many $\chi$ (real-valued as above) for which
$L(s,\chi)$ has no critical points on the interval $1 < s < \infty$ (i.e. monotonic)?
2) Is it possible for $L(s,\chi)$ to have more than one critical point in $(1,\infty)$? If so, is there any upper bound on the number of such critical points which holds for all $\chi$?
3) Are there any books or papers which display graphs of $L$-functions on the real line?
Thanks

REPLY [8 votes]: I can answer 2): if q=857, the L-function has 2 critical points. I would guess
the number of critical points is unbounded. I do not know
if this is the smallest example (draw on the same graph the L-function
for q=17 and for q=857 starting at t=2, say (the critical points are smaller):
the graphs almost coincide; Andrew Granville confirms that this is is a case
of what he calls "pretentious characters", this is why I tried 857 immediately.
UPDATE: I have run a program for $q$ from $-10000$ to $10000$: there are 27 positive
q and 27 negative q for which there are 2 critical points (unfortunately none
with more than 2), the first positive $q$ being 593, 857, 1697,..., and the
first negative $q$ being $-548$, $-632$, $-872$, ...
I should also have answered question 3): in Pari/GP the command
L=lfuninit(D,[5,5,1]);ploth(t=1,10,lfun(L,t))
plots the $L$-function of a real character $\chi_D$ from $1$ to $10$ for instance.