Different Recurrent Neural Network (RNN) architectures update their state in different manners as the input sequence is processed. RNNs including a multiplicative interaction between their current state and the current input, second-order ones, show promising performance in language modeling. In this paper, we introduce a second-order RNNs that generalizes existing ones. Evaluating on the Penn Treebank dataset, we analyze how its different components affect its performance in character-lever recurrent language modeling. We perform our experiments controlling the parameter counts of models. We find that removing the first-order terms does not hinder performance. We perform further experiments comparing the effects of the relative size of the state space and the multiplicative interaction space on performance. Our expectation was that a larger states would benefit language models built on longer documents, and larger multiplicative interaction states would benefit ones built on larger input spaces. However, our results suggest that this is not the case and the optimal relative size is the same for both document tokenizations used.