We examine the applicability of modern neural network architectures to the midterm prediction of earthquakes. Our data-based classification model aims to predict if an earthquake with the magnitude above a threshold takes place at a given area of size $10 \times 10$ kilometers in $10$-$60$ days from a given moment. Our deep neural network model has a recurrent part (LSTM) that accounts for time dependencies between earthquakes and a convolutional part that accounts for spatial dependencies. Obtained results show that neural networks-based models beat baseline feature-based models that also account for spatio-temporal dependencies between different earthquakes. For historical data on Japan earthquakes our model predicts occurrence of an earthquake in $10$ to $60$ days from a given moment with magnitude $M_c > 5$ with quality metrics ROC AUC $0.975$ and PR AUC $0.0890$, making $1.18 \cdot 10^3$ correct predictions, while missing $2.09 \cdot 10^3$ earthquakes and making $192 \cdot 10^3$ false alarms. The baseline approach has similar ROC AUC $0.992$, number of correct predictions $1.19 \cdot 10^3$, and missing $2.07 \cdot 10^3$ earthquakes, but significantly worse PR AUC $0.00911$, and number of false alarms $1004 \cdot 10^3$.