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Context. About 120 Be/X-ray binaries (BeXBs) are known in the SmallMagellanic Cloud (SMC); about half of them are pulsating with periods from afew to hundreds of seconds. SXP 1323 is one of the longest-period pulsars knownin this galaxy. Aims. SXP 1323 is in the field of view of a large set ofcalibration observations that we analyse systematically, focusing on the timeanalysis, in search of periodic signals. Methods. We analyse all availableX-ray observations of SXP 1323 from Suzaku, XMM-Newton, and Chandra, in thetime range from 1999 to the end of 2016. We perform a Lomb-Scargle periodogramsearch in the band 2.5-10 keV on all observations to detect the neutron starspin period and constrain its long-term evolution. We also perform an orbitalperiod search on the long-term light curve, merging all datasets. Results. Wereport the discovery of a 26.188+-0.045 d period analysing data from Suzaku,XMM-Newton, and Chandra, which confirms the optical period derived from theOptical Gravitational Lensing Experiment (OGLE) data. If this corresponds tothe orbital period, this would be very short with respect to what is expectedfrom the spin/orbital period relationship. We furthermore report on the spinperiod evolution in the last years. The source is spinning-up with an averagerate of Pdot/P of 0.018 yr-1, decreasing from 1340 to 1100 s, in the periodfrom 2006 to the end of 2016, which is also extreme with respect to the otherBe/X-ray pulsars. From 2010 to the end of 2014, the pulse period is not clearlydetectable, although the source was still bright. Conclusions. SXP 1323 is apeculiar BeXB due to its long pulse period, rapid spin-up for several years,and short orbital period. A continuous monitoring of the source in the nextyears is necessary to establish the long-term behaviour of the spin period.

Discovery of a 26.2 day period in the long-term X-ray light curve of SXP 1323: a very short orbital period for a long spin period pulsar