Genome modeling conventionally treats gene sequence as a language, reflecting its structured motifs and long-range dependencies analogous to linguistic units and organization principles such as words and syntax. Recent studies utilize advanced neural networks, ranging from convolutional and recurrent models to Transformer-based models, to capture contextual information of gene sequence, with the primary goal of obtaining effective gene sequence representations and thus enhance the models' understanding of various running gene samples. However, these approaches often directly apply language modeling techniques to gene sequences and do not fully consider the intrinsic information organization in them, where they do not consider how units at different granularities contribute to representation. In this paper, we propose DNAZEN, an enhanced genomic representation framework designed to learn from various granularities in gene sequences, including small polymers and G-grams that are combinations of several contiguous polymers. Specifically, we extract the G-grams from large-scale genomic corpora through an unsupervised approach to construct the G-gram vocabulary, which is used to provide G-grams in the learning process of DNA sequences through dynamically matching from running gene samples. A Transformer-based G-gram encoder is also proposed and the matched G-grams are fed into it to compute their representations and integrated into the encoder for basic unit (E4BU), which is responsible for encoding small units and maintaining the learning and inference process. To further enhance the learning process, we propose whole G-gram masking to train DNAZEN, where the model largely favors the selection of each entire G-gram to mask rather than an ordinary masking mechanism performed on basic units. Experiments on benchmark datasets demonstrate the effectiveness of DNAZEN on various downstream tasks.