14.2. The GL(2) setting
Fix a normalised cuspidal Hecke eigenform f of weight k+2 and level
\Gamma_0(N) with p \mid N, and let L(f,s) be its complex
L-function. There are again three ways to attach a p-adic L-function to
f: analytic, arithmetic and algebraic. We treat each in turn, and then state
the comparison theorems and the Main Conjecture for f.
A complex argument s = j is critical for L(f,s) in the sense of
Deligne: j is critical for a motivic L-function L(M,s) when neither the
Euler factor at infinity L_\infty(M,s) nor L_\infty(M,1-s) has a pole at
j. For the eigenform f this criterion singles out the values
L(f,\chi,j+1) for \chi an arbitrary Dirichlet character and
0 \le j \le k. (For \GL(1), the same criterion recovers the negative odd
integers and positive even integers as the critical values of \zeta(s),
matching the Kubota–Leopoldt interpolation range.)