Nonvanishing of Dirichlet L-functions at the central point

Let $\chi$ be a Dirichlet character and let $L(s,\chi)$ denote its Dirichlet $L$-function. The central-point nonvanishing conjecture.

$$L\left(\frac12,\chi\right)\ne 0$$

for every Dirichlet character $\chi$. The source presents this as a consequence of the linear-independence conjecture, since any set containing $0$ is linearly dependent. Partial results establish nonvanishing for positive proportions or infinite families of characters, but the universal assertion remains open.