# Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s}$$ to be the $L$-function.

Further assume that $f$ is an eigenfunction of the half integral weight Hecke operators.

Has there been located any zeros of this $L$ function, for any choice of $f$, in the critical strip which are not on the critical line $\Re(s) = \frac 12$.

• Have you done any computations yourself? While I'm dubious that this should be true for almost any form, it's worth noting that $L(s,\theta_\chi)=L(2s-1/2,\chi)$ for a non-trivial Dirichlet character $\chi$, so RH presumably holds in this case. In general, though, the multiplicative structure of half-integral weight eigenforms is more complex, and I'd be very surprised if it were to hold if the form is orthogonal to the space of unary theta functions. – rlo Apr 17 '13 at 3:08
• No I haven't done any computations, but how would one go about computing the zeros of such a modular form. I agree that for theta functions something special must be happening. – Eren Mehmet Kiral Apr 19 '13 at 21:40

## 1 Answer

The following paper (page 6) discuss some examples of half-integral weight $L$-functions which has zeros not necessarily on the critical line.

https://arxiv.org/pdf/math/9411213.pdf

It is published in J. Math. Kyoto Univ.(1995),663-696. This reference may be helpful.

• As far as I understand the L functions on page 6 are symmetric power L functions (so they are higher degree but not half integral weight). The fact that the zeros are not on the 1/2 line (but on some other vertical line) has to be an issue about normalization. It is rare that people call what I have defined above an L function, it is certainly a Dirichlet series with functional equation. – Eren Mehmet Kiral Oct 27 '17 at 20:35
• Thanks for your attention. The section of modular forms of half-integral weight starts from the last line of page 6 and the detail is given page 7 onwards. In this section the author considers Dirichlet series attached to forms of integral weight and half-integral weight and one by one examine their zeros. – Karam Deo Shankhadhar Oct 28 '17 at 9:26