/*** Dedekind zeta-function of a number field F/Q ***/ /*** v1.3, December 2013, questions to [email protected] ***/ /*** ***/ /*** type \rex-nf or read("ex-nf") at pari prompt to run this ***/ fpol = x^4-x^2-1; \\ polynomial which defines F/Q \\ may change this to any other polynomial read("computel"); \\ read the ComputeL package \\ and set the default values default(realprecision,20); \\ set working precision; used throughout nf = nfinit(fpol); \\ initialize the number field F/Q zinit = zetakinit(nf); \\ and the built-in Dedekind zeta-function LPari(x) = zetak(zinit,x); print("EXAMPLE: Dedekind zeta-function L(s)=zeta_F(s) of number field F"); print(" with ",default(realprecision)," digits precision"); print("F = Q(root of ",fpol,")"); print("[F:Q] = ",Fdegree = poldegree(fpol)); print("Discriminant = ",disc = nf.disc); print("r1 (real emb.) = ",r1 = nf.sign[1]); print("r2 (complex emb.) = ",r2 = nf.sign[2]); print("Gamma factor = ",gammaV = concat(vector(r1+r2,X,0),vector(r2,X,1))); conductor = abs(disc); \\ exponential factor weight = 1; \\ L(s)=sgn*L(weight-s) sgn = 1; \\ sign in the functional equation Lpoles = [1]; \\ pole at s=1, Lresidues=automatic dzk = dirzetak(nf,cflength()); \\ coefficients a(k) in L(s) initLdata("dzk[k]"); \\ initialize L-series \\ Determine residue at s=1 and check the functional equation print("Error in func. eq. = ",errprint(checkfeq())); \\ Compare L(2) and built-in LPari(2) print("L(2) = ",L(2)); print(" (or using pari) = ",LPari(2)); \\ Compare residue at 1 and the one given by class number formula print("Residue at s=1"); print(" (automatically determined) = ",Lresidues[1]); \\ Determine the residue at s=1 using the class number formula bnf = bnfinit(fpol); residue = - 2^(r1+r2) * Pi^(r2/2) * bnf.reg * bnf.clgp.no / bnf.tu[1]; print(" (class number formula) = ",residue);
