Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with spectral parameters $\{\mu_j\}$ and $\{\nu_k\}$. We say that $L(s)$ is of algebraic type if either every $\mu_j$ and $\nu_k$ is in $\mathbb{Z}$, or every $\mu_j$ and $\nu_k$ is in $\frac{1}{2}+\mathbb{Z}$. The integer $w_{alg}=2\max\{0,\nu_1,\dots,\nu_{d_2}\}$ is called the algebraic weight of the $L$-function.
Definition 4.2. Suppose $L(s) = \sum a_n n^{-s}$ is an analytic $L$-function. We say that $L(s)$ is of arithmetic type if there exists $w_{ar}\in\mathbb{Z}$ and a number field $F$ such that $a_nn^{w_{ar}/2}\in\mathcal{O}_F$ for all $n$. The smallest such $F$ is called the field of coefficients, and the smallest such $w_{ar}$ is called the arithmetic weight of the $L$-function.
The statement I am curious to know more about is the following given at the beginning of page 13 of the linked paper:
Furthermore, we have the Hodge conjecture: $w_{alg} = w_{ar}$.
Firstly, is this the same Hodge conjecture as the usual millennium problem asking whether the dimension of the space of algebraic cycles equals the dimension of a cohomology group? If so, how does one explain this supposed connection between the usual Hodge conjecture and $L$-functions?
My rough understanding of the equivalence between this statement and the usual way the Hodge conjecture is stated is the following:
The unresolved Hodge conjecture is about algebraic varieties and any algebraic variety $X$ does indeed have an $L$-function. The values of the $L$-function at certain points along the real axis (related to algebraic weight) appear to capture information about the dimension of cohomology groups of $X$. In contrast, the coefficients of the Dirichlet series (related to arithmetic weight) appear to capture information about algebraic subvarieties of $X$. So to say that arithmetic and algebraic weights are equal might amount to saying that the dimension of the space of algebraic cycles equals the dimension of a cohomology group?
The relationship between $w_{alg}$ and cohomology appears potentially easier to explain: for any variety, the motive computing its $k$-th cohomology has weight $k$, and hence, per the claim of the authors of the linked paper in section 4.1, the attached $L$-function will also have weight $k$. However, as far as I am aware, this says nothing about the dimension of this cohomology space, let alone the space of Hodge cycles inside of it.
For context, I became aware of this equivalence from this great YouTube video, however, the relationship to the Hodge conjecture seems to also be unknown to the creator of the video, and thus I was curious to find out more as it would seem to suggest that all 3 unresolved millennium problems about pure mathematics (not considering P vs. NP) are in some way related to $L$-functions.