Riemann's zeta-function satisfies a convexity bound on the critical line; moreover, $\zeta(s)$ is non-vanishing in a neighborhood of the edge of the critical strip. Let $L(s,\pi,r)$ be a Langlands-Shahidi L-function; what generalizations of $\zeta(s)$ can be said about this $L(s,\pi,r)$? Does it satisfy a convexity bound on the critical line? Is it non-vanishing close to the edge of the critical strip? ...