# Zeros of L function on the 0.5 line

Could someone please tell what results are known about the zeros of L function, $$L(s,\chi)$$ on the 1/2 line, where $$\chi$$ is a character mod $$q$$? Is there an upper bound for this count when we count zeros with $$|\Im s|\leq T?$$

In particular, what is known about zeta function? I know there are some explicit version of Von Mangoldt theorem that have been done for Zeta function. I want to know what are some relevant results for any $$L(s,\chi).$$

I guess there should be some results available for general $$L$$ functions, but I am unable to find them myself.

Any help would be appreciated. Thanks in advance.

Added: I meant an explicit formula for upper bound when I asked for one.

• Concerning explicit formulae you may be interested by this answer (with explicit results for $q=4$). Commented Mar 24 at 19:09

Let's state a general zero-counting form for a general $$L$$-function. Take an $$L$$-function with the functional equation

$$\Lambda(s) := L(s) Q^s \prod_{\nu = 1}^N \Gamma( \alpha_\nu + \beta_\nu ) = \omega \overline{\Lambda(1 - \overline{s})}.$$

Define the degree of $$L$$ to be

$$d_L := 2 \sum_{\nu} \alpha_\nu.$$

We also define the convenience variable

$$\alpha := \prod_\nu \alpha_\nu^{2 \alpha_\nu}.$$

In practice, $$d_L$$ is an integer for natural $$L$$-functions. It is $$1$$ for $$\zeta(s)$$ and Dirichlet $$L$$-functions $$L(s, \chi)$$.

Then the number of nontrivial zeros of $$L(s)$$ of the form $$\rho = \sigma + i t$$ with $$\lvert t \rvert < T$$ is given by

$$N(T) = \frac{d_L}{2 \pi} T \log \frac{4T}{e} + \frac{T}{2 \pi} \log(\alpha Q^2) + O(\log T).$$

We expect all of these to be on the $$1/2$$ line, but we don't know that. In many (maybe all?) cases we know at least that a positive proportion of the zeros are are the $$1/2$$ line. For degree $$d_L \leq 2$$, we also know that $$100$$ percent of zeros are within $$\epsilon$$ of the $$1/2$$ line for any $$\epsilon$$.

The $$N(T)$$ asymptotic is well-known and can be found (or quickly derived) from standard analytic number theory texts such as Montgomery and Vaughan, Iwaniec and Kowalski, or the classic books on $$\zeta(s)$$ by Heath-Brown or Titchmarsh.

Proving that there is a positive proportion of zeros on the line is also classic and also in Montgomery and Vaughan or Titchmarsh, at least for $$\zeta(s)$$. I don't know if anyone has written this down for $$L(s, \chi)$$, but it should be substantially similar.

Broadly, if you have questions about how someone proved something about $$\zeta(s)$$, it's a good idea to look through Titchmarsh's books on $$\zeta$$. There is an enormous wealth of techniques.

• the only comment I would add here is that if the character is not primitive, the corresponding $L$ function usually has tons of zeroes on $\Re s =0$, and while you may consider those to be "trivial" of course, they tend to change the look of the functional equation and so on so usually one assumes primitive characters when making analogies with $\zeta$ Commented Mar 6 at 15:33
• Thanks for your answer. But I asked for explicit formulas. Please check the new edit in the question. Commented Mar 6 at 16:52