Moments and One level density of sextic Hecke $L$-functions

The following notations and conventions are used throughout the paper.
We write $\Phi(t)$ for $\Phi_{X}(t)$.
$e(z)=\exp(2\pi iz)=e^{2\pi iz}$.
$f=O(g)$ or $f\ll g$ means $|f|\leq cg$ for some unspecified positive constant $c$.
$f=o(g)$ means $\lim_{x\to\infty}f(x)/g(x)=0$.
$\mu_{[\omega]}$ denotes the Möbius function on $\mathbb{Z}[\omega]$.
$\zeta_{\mathbb{Q}(\omega)}(s)$ stands for the Dedekind zeta function of $\mathbb{Q}(\omega)$.

It is well-known that $K=\mathbb{Q}(\omega)$ has class number $1$. For $j=2,3,6$, the symbol $(\frac{\cdot}{n})_{j}$ is the quadratic ($j=2$), cubic ($j=3$) and sextic ($j=6$) residue symbol in the ring of integers $\mathcal{O}_{K}=\mathbb{Z}[\omega]$. For a prime $\varpi\in\mathbb{Z}[\omega],(\varpi,j)=1$, we define for $a\in\mathbb{Z}[\omega]$, $(a,\varpi)=1$ by $\left(\frac{a}{\varpi}\right)_{j}\equiv a^{(N(\varpi)-1)/j}\pmod{\varpi}$, with $\left(\frac{a}{\varpi}\right)_{j}\in<(-\omega)^{6/j}>$, the cyclic group of order $j$ generated by $(-\omega)^{6/j}$. When $\varpi|a$, we define $\left(\frac{a}{\varpi}\right)_{j}=0$. Then these symbols can be extended to any composite $n$ with $(N(n),j)=1$ multiplicatively. We further define $\left(\frac{\cdot}{n}\right)_{j}=1$ when $n$ is a unit in $\mathbb{Z}[\omega]$. We note that we have $\left(\frac{\cdot}{n}\right)^{2}_{6}=\left(\frac{\cdot}{n}\right)_{3},\left(\frac{\cdot}{n}\right)^{3}_{6}=\left(\frac{\cdot}{n}\right)_{2}$ whenever $\left(\frac{\cdot}{n}\right)_{6}$ is defined.

We say that $n=a+b\omega$ in $\mathbb{Z}[\omega]$ is primary if $n\equiv\pm 1\pmod{3}$, which is equivalent to $a\not\equiv 0\pmod{3}$, and $b\equiv 0\pmod{3}$ (see [B&Y, p. 209]).

Recall that (see [B&Y, p. 883]) the following cubic reciprocity law holds for two co-prime primary $n,m$ :

We also have the following supplementary laws for a primary $n=a+b\omega,n\equiv 1\pmod{3}$ (see [Lemmermeyer, Theorem 7.8]),

Following the notations in [Lemmermeyer, Section 7.3], we say that any primary $n=a+b\omega\in\mathbb{Z}[\omega]$, with $(n,6)=1$ is $E$-primary if

It follows from [Lemmermeyer, Lemma 7.9] that $n$ is $E$-primary if and only if $n^{3}=c+d\omega$ with $c,d\in\mathbb{Z}$ such that $6|d$ and $c+d\equiv 1\pmod{4}$. This implies that products of $E$-primary numbers are again $E$-primary. Note that in $\mathbb{Z}[\omega]$, every ideal co-prime to $6$ has a unique $E$-primary generator. Furthermore, the following sextic reciprocity law holds for two $E$-primary, co-prime numbers $n,m\in\mathbb{Z}[\omega]$ :

We also have the following supplementary laws for an $E$-primary $n=a+b\omega$ with $(n,6)=1$ (see [Lemmermeyer, Theorem 7.10]):

where $\left(\frac{\cdot}{\cdot}\right)_{\mathbb{Z}}$ denotes the Jacobi symbol in $\mathbb{Z}$.

The above discussions allow us to define a sextic Dirichlet character $\chi^{(72c)}\pmod{72c}$ for any element $c\in\mathcal{O}_{K},(c,6)=1$, such that for any $n\in(\mathcal{O}_{K}/72c\mathcal{O}_{K})^{*}$,

One deduces from (2.1), (2.2) and the sextic reciprocity that $\chi^{(72c)}(n)=1$ when $n\equiv 1\pmod{72c}$. It follows from this that $\chi^{(72c)}(n)$ is well-defined. As $\chi^{(72c)}(n)$ is clearly multiplicative, of order $6$ and trivial on units, it can be regarded as a sextic Hecke character modulo $72c$ of trivial infinite type. We write $\chi^{(72c)}$ for this Hecke character as well and we call it the Kronecker symbol. Furthermore, if $c$ is square-free, $\chi^{(72c)}$ is non-principal and primitive. To see this, we write $c=u_{c}\cdot\varpi_{1}\cdots\varpi_{k}$ with a unit $u_{c}$ and primes $\varpi_{j}$. Suppose $\chi^{(72c)}$ is induced by some $\chi$ modulo $c^{\prime}$ with $\varpi_{j}\nmid c^{\prime}$, then by the Chinese Remainder Theorem, there exists an $n$ such that $n\equiv 1\pmod{72c/\varpi_{j}}$ and $\left(\frac{\varpi_{j}}{n}\right)_{6}\neq 1$. It follows from this that $\chi(n)=1$ but $\chi^{(72c)}(n)\neq 1$, a contradiction. Thus, $\chi^{(72c)}$ can only be possibly induced by some $\chi$ modulo $36c$ or modulo $8(1-\omega)^{3}c$. Suppose it is induced by some $\chi$ modulo $36c$, then by the Chinese Remainder Theorem, there exists an $n$ such that $n\equiv 1\pmod{9c}$ and $n\equiv 1+4\omega\pmod{8}$, then we have $\chi(n)=1$ but $\chi^{(72c)}(n)=\left(\frac{2}{n}\right)_{2}\neq 1$ by (2.2), a contradiction. Suppose it is induced by some $\chi$ modulo $8(1-\omega)^{3}c$, then by the Chinese Remainder Theorem, there exists an $n$ such that $n\equiv 1\pmod{8c}$ and $n\equiv 1+3(1-\omega)\pmod{9}$, then we have $\chi(n)=1$ but

by (2.1) (note that we have $3=-\omega^{2}(1-\omega)^{2}$), a contradiction. This implies that $\chi^{(72c)}$ is primitive. This also shows that $\chi^{(-72c)}$ is non-principal.

Suppose that $n\in\mathbb{Z}[\omega]$ with $(n,6)=1$. For $j=2,3,6$, the quadratic ($j=2$), cubic ($j=3$) and sextic ($j=6$) Gauss sum is defined by

where $\widetilde{e}(z)=e\left((z-\overline{z})/\sqrt{-3}\right)$.

We can infer from its definition that $g_{j}(1)=1$. Moreover, the following well-known relation (see [P, p. 195]) holds for all $n$:

More generally, for any $n,r\in\mathbb{Z}[\omega]$, with $(n,6)=1$, we set

We need the following properties of $g_{j}(r,n)$:

Let $c\in\mathcal{O}_{K},c\equiv 1\pmod{36}$ be square-free. It follows from (2.2) that $\chi_{c}=\left(\frac{\cdot}{c}\right)_{6}$ is trivial on units, it can be regarded as a primitive Hecke character $\pmod{c}$ of trivial infinite type. The Hecke $L$-function associated with $\chi_{c}$ is defined for $\Re(s)>1$ by

where $\mathcal{A}$ runs over all non-zero integral ideals in $K$ and $N(\mathcal{A})$ is the norm of $\mathcal{A}$. As shown by E. Hecke, $L(s,\chi_{c})$ admits analytic continuation to an entire function and satisfies a functional equation. We refer the reader to [Gu, Luo, G&Zhao1, G&Zhao20] for a more detailed discussion of these Hecke characters and $L$-functions.

Let $G(s)$ be any even function which is holomorphic and bounded in the strip $-4<\Re(s)<4$ satisfying $G(0)=1$. We have the following expression for $L(1/2+it,\chi_{c})$ for $t\in\mathbb{R}$ (see [G&Zhao20, Section 2.4]):

where $g_{6}(c)$ is the Gauss sum defined in Section 2.1, $D_{K}=-3$ is the discriminant of $K$ and

We write $V$ for $V_{0}$ and note that for a suitable $G(s)$ (for example $G(s)=e^{s^{2}}$), we have for any $c>0$ (see [HIEK, Proposition 5.4]):

On the other hand, when $G(s)=1$, we have (see [sound1, Lemma 2.1]) for the $j$-th derivative of $V(\xi)$,

For any Hecke character $\chi\pmod{36}$ of trivial infinite type, we let

The following lemma gives the analytic behavior of $h(r,s;\chi)$ on $\Re(s)>1$.

Lemma 2.6 is shown in the same manner as that of [G&Zhao1, Lemma 2.5]. We use Lemma 2.7 to remove the condition $(n,r)=1$ in (2.10), and the result of Lemma 2.6 then follows from the proof of the Lemma on [P, p. 200], taking into account the observation from (2.6) that $|g_{6}(\varpi^{k},\varpi^{l})|\leq N(\varpi)^{k+1/2}$ when $k<l$ for any $E$-primary prime $\varpi$.

To state the next lemma, we define for $a,c\in\mathbb{Z}[\omega],(ac,6)=1$,

It is easy to see that $\psi_{a}(c)$ is a Hecke character $\pmod{36}$ of trivial infinite type.

Furthermore, for any square-free $a\in\mathbb{Z}[\omega]$, we let $\{\varpi_{1},\cdots,\varpi_{k}\}$ be the set of distinct $E$-primary prime divisors of $a$ and we define for $1\leq j\leq 4$,

where we set the empty product to be $1$. In particular, we have $P_{j}(a)=1$ when $a$ is a unit and $\Psi_{j}(a)=1$ when $a$ is a unit or a prime. As $\left(\frac{\varpi^{j}_{2}}{\varpi^{j+1}_{1}}\right)_{6}=\left(\frac{\varpi_{2}}{\varpi_{1}}\right)^{j(j+1)}_{6}$ for two distinct $E$-primary primes $\varpi_{1},\varpi_{2}$, one checks easily by induction on the number of prime divisors of $a$ and the sextic reciprocity that $P_{j}(a)$ is independent of the order of $\{\varpi_{1},\cdots,\varpi_{k}\}$ when $j\neq 3$. Similarly, as $\left(\frac{\varpi_{1}}{\varpi_{2}}\right)_{3}=\left(\frac{\varpi_{2}}{\varpi_{1}}\right)_{3},\psi_{\varpi_{1}}(\varpi_{2})=\psi_{\varpi_{2}}(\varpi_{1})$, $P_{3}(a)$ and $\Psi_{j}(a),1\leq j\leq 4$ are also independent of the order of $\{\varpi_{1},\cdots,\varpi_{k}\}$.

Now we have