Special values of symmetric power $L$-functions and Hecke eigenvalues

Details Special values of symmetric power $L$-functions and Hecke eigenvalues Special values of symmetric power $L$-functions and Hecke eigenvalues

2006-09-08

arxiv.org/abs/math/0609227v2

Mathematics

Number Theory

42 pages Previously circulated under the title "Central values and values at the edge of the critical strip of symmetric power

Scientific paper

We compute the moments of L-functions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the L-functions of modular forms. We show that, in the case of even powers, it is equivalent to twist by the value at the edge of the critical strip of the symmetric square L-functions. We deduce information on the size of symmetric power L-functions at the edge of the critical strip under conditions. In a second part, we study the distribution of small and large Hecke eigenvalues. We deduce information on the simultaneous extremality conditions on the values of L-functions of symmetric powers of modular forms at the edge of the critical strip.

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