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In 1994, van Geemen and Top constructed a non-selfdual motive of rank threeover $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdualautomorphic representation of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$ of level$\Gamma_0(128)$. They experimentally confirmed the coincidence of the local$L$-factors at finitely many primes using computer. In this paper, we shallprove the coincidence of the local $L$-factors at every prime. To show this, weuse the recent results of Harris-Lan-Taylor-Thorne and Scholze on theconstruction of Galois representations, and Greni\'e's results to comparethree-dimensional $2$-adic Galois representations. We also prove thelocal-global compatibility at $p = 2$, including the case $p = \ell$.

Galois representations associated with a non-selfdual automorphic representation of GL(3)