Chief Minister Siddaramaiah announced this at a Press conference after holding a high-level meeting with police officers on Wednesday.
"The silencing of a journalist in this manner has risky portents for democracy", the Indian Women's Press Corps said in a statement.
Several social organisations, writers and journalists unions have strongly condemned the assassination of the 55-year-old journalist at her residence in Bengaluru on Tuesday night by unidentified assailants, who fired at her at point-blank range.
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She was laid to rest with full state honours amid chants of "Gauri Lankesh Amar Rahe (Long live Gauri Lankesh)" as her body was lowered into the grave. Also, he said his Government has an "open mind" to an investigation by the CBI, a demand made by the slain journalist's family.
"Law and order of Karnataka is the responsibility of the state government, under Congress".
Investigators pinned their hopes on the CCTV footage to get some clues about the identity of the killers.
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In a Facebook post shared Saturday, Edward Burleigh called his daughter a "beautiful angel" who is now in heaven. Police say Joshua Hupperterz, 29, admitted to 'elements of the crime, ' as indicated by the New York Daily News .
The BJP today criticised Rahul Gandhi for linking Gauri Lankesh's murder to the RSS-BJP and said neither the party nor any organisation linked to it was involved in the crime.
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Police said they were searching for leads, but that it was too early to say who killed her.
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He termed Rahul's comments against Prime Minister Narendra Modi "unfortunate".