We report on a search for Fast Radio Bursts (FRBs) with the Green Bank Northern Celestial Cap (GBNCC) Pulsar Survey at 350 MHz. Pointings amounting to a total on-sky time of 61 days were searched to a DM of 3000 pc cm$^{-3}$ while the rest (23 days; 29% of the total time) were searched to a DM of 500 pc cm$^{-3}$. No FRBs were detected in the pointings observed through May 2016. We estimate a 95% confidence upper limit on the FRB rate of $3.6times 10^3$ FRBs sky$^{-1}$ day$^{-1}$ above a peak flux density of 0.63 Jy at 350 MHz for an intrinsic pulse width of 5 ms. We place constraints on the spectral index $alpha$ by running simulations for different astrophysical scenarios and cumulative flux density distributions. The non-detection with GBNCC is consistent with the 1.4-GHz rate reported for the Parkes surveys for $alpha > +0.35 $ in the absence of scattering and free-free absorption and $alpha > -0.3$ in the presence of scattering, for a Euclidean flux distribution. The constraints imply that FRBs exhibit either a flat spectrum or a spectral turnover at frequencies above 400 MHz. These constraints also allow estimation of the number of bursts that can be detected with current and upcoming surveys. We predict that CHIME may detect anywhere from several to $sim$50 FRBs a day (depending on model assumptions), making it well suited for interesting constraints on spectral index, the log $N$-log $S$ slope and pulse profile evolution across its bandwidth (400-800 MHz).
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