Decompositions of the extended Selberg class functions

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> F\left(s) be a function from the extended Selberg class. We consider decompositions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> F\left(s)=f\left(h\left(s)) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>h</m:mi> </m:math> h are meromorphic functions. Among other things, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F is prime if and only if the greatest common divisor of the orders of all zeros and the pole of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F is 1.