  The reported association of high-energy neutrino event IceCube-170922A and
blazar TXS 0506+056 has sparkled discussion about blazars as sources of cosmic
neutrinos. In this paper, we use publicly released IceCube data and blazar
locations from Roma-BzCat to test spacial correlation between neutrino events
and blazar locations. We also scrutinize the correlation between $\gamma$-ray
flux and neutrino flux of blazars by applying a temporal filter onto the data
based on Fermi monitored source list. We find no compelling evidence to prove
blazars as the main source of cosmic neutrinos, as known before the detected
IceCube-170922A / TXS 0506+056 association. While we do not rule out the
association between IceCube-170922A and TXS 0506+056, the significance level we
obtained is not high enough to claim a strong association. If such an
association is real, a special physical condition is desired to allow a small
fraction of blazars to become bright neutrino sources.
