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?


  • $\begingroup$ Is it anyhow related to number of sign changes of $ \chi_{q}(n) $ in the range $ n\in(1,q)\cap\mathbb{N} $? $\endgroup$ Mar 18, 2018 at 21:34
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    $\begingroup$ There is a large literature on L-functions of graphs, but graphs of L-functions, not so much. $\endgroup$ Mar 18, 2018 at 22:11
  • 1
    $\begingroup$ arxiv.org/abs/1304.0827 seems to be tangentially related. $\endgroup$ Mar 18, 2018 at 22:23

1 Answer 1


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


plots the $L$-function of a real character $\chi_D$ from $1$ to $10$ for instance.


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