The incident occurred in Durham in England on Saturday evening, The Northern Echo reported.
The 13-year-old was arrested on suspicion of animal cruelty and causing criminal damage but was since released under investigation.
It is believed an air rifle was used in the incident. A spokesperson said that the police are currently in attempting to locate two more youth in relation to the incident.
Inspector Lee Blakelock also condemned the incident.
“Several animals have been injured with one fatally after being shot in the head with what is believed to be an air weapon,” he said.
“This calf has had to be put down by a vet,” he concluded.
The investigation is currently ongoing.
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