Breakthrough Listen (BL) is a ten-year initiative to search for signatures of technologically capable life beyond Earth via radio and optical observations of the local Universe. A core part of the BL program is a comprehensive survey of 1702 nearby stars at radio wavelengths (1-10 GHz). Here, we report on observations with the 64-m CSIRO Parkes radio telescope in New South Wales, Australia, and the 100-m Robert C. Byrd Green Bank radio telescope in West Virginia, USA. Over 2016 January to 2019 March, a sample of 1138 stars was observed at Green Bank using the 1.10-1.90 GHz and 1.80-2.80 GHz receivers, and 189 stars were observed with Parkes over 2.60-3.45 GHz. We searched these data for the presence of engineered signals with Doppler-acceleration drift rates between -4 to 4 Hz/s. Here, we detail our data analysis techniques and provide examples of detected events. After excluding events with characteristics consistent with terrestrial radio interference, we are left with zero candidates. Given the sensitivity of our observations, we can put an upper limit on the power of potential radio transmitters at these frequencies at 2x10^12 W, and 9x10^12 W for GBT and Parkes respectively. These observations constitute the most comprehensive search over 1.10-3.45 GHz for technosignatures to date. All data products, totalling ~219 TB, are available for download as part of the first BL data release (DR1), as described in a companion paper (Lebofsky et. al., 2019)
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