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Neural Audio Codecs (NACs) have gained growing attention in recent years as technologies for audio compression and audio representation in speech language models. While mainstream NACs typically require G-level computation and M-level parameters, the performance of lightweight and streaming NACs remains underexplored. This paper proposes SpecTokenizer, a lightweight streaming codec that operates in the compressed spectral domain. Composed solely of alternating CNN and RNN layers, SpecTokenizer achieves greater efficiency and better representational capability through multi-scale modeling in the compressed spectrum domain. At 4 kbps, the proposed SpecTokenizer achieves comparable or superior performance compared to the codec with state-of-the-art lightweight architecture while requiring only 20% of the computation and 10% of the parameters. Furthermore, it significantly outperforms the codec when using similar computational and storage resources.