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Two types of self-similarity exhibited in 1st degree L-functions on the line S=1+i*T are examined in therange (10 5 < T < 10 30 ). Firstly, it is observed that there is extended mesoscale structure surrounding thelarge peaks of the L-function on the line S=1+i*T reflecting closely the product function of (i) a truncatedRiemann Zeta function near the real axis with (ii) simple Euler factor type terms arising from the absentlower modulo primes of the L-function. A second self-similarity patterndoi:10.6084/m9.figshare.13242101.v4 fatcat:3os6sakeofefxppnmwvzatg76a