/*** EXAMPLE: Shintani's zeta-function ***/ /*** v1.3, December 2013, questions to tim.dokchitser@bristol.ac.uk ***/ /*** ***/ /*** type \rex-shin or read("ex-shin") at pari prompt to run this ***/ read("computel"); \\ read the ComputeL package \\ and set the default values default(realprecision,19); \\ set working precision; used throughout \\ larger precision needs more coefficients \\ initialize L-function parameters GlobalC = 1/2/sqrt(Pi); \\ Global coefficient in front of L(s) conductor = 432; \\ exponential factor gammaV = [0,1,-1/6,1/6]; \\ list of gamma-factors weight = 1; \\ L(s)=sgn*L(weight-s) sgn = 1; \\ sign in the functional equation Lpoles = [1/6,1]; \\ ``half'' of the poles of L(s) {Lresidues = \\ and residues in there [zeta(2/3)*2^(3/2)*3^(-1/4)*Pi^(-1/3)*gamma(1/3)/GlobalC, \\ in s=1/6 gamma(1/12)*gamma(-1/12)/sqrt(3)/12/GlobalC];} \\ in s=1 \\ note: can also set Lresidues=automatic, checkfeq() \\ will then determine them to decent precision \\ First 400 Coefficients {c1=[1/3,1,1,1,1,1,1,4/3,1,1,1,1,1,1,1,2,1,3,1,1,1,1,1,2,5/3,1,1,1,1,1,1,2,1,3, 1,1,1,1,1,2,2,1,1,1,1,1,1,2,1,3,1,1,1,2,1,2,1,1,3,1,1,1,2,2,1,3,1,1,1,1,1, 4,1,3,1,1,1,1,1,2,1,3,1,1,5/3,1,1,2,1,1,1,1,1,1,2,2,1,3,1,1,1,1,1,2,1,1,1, 2,1,1,1,2,3,3,3,1,1,1,1,2,1,1,1,1,1,3,1,10/3,3,3,1,1,1,1,1,4,1,1,1,1,1,1, 1,2,1,3,1,1,2,1,1,2,1,1,1,1,1,3,1,4,3,4,1,1,1,1,3,2,1,1,1,1,1,1,1,2,1,3,1, 3,5/3,1,1,2,1,1,1,1,1,1,1,2,1,3,1,3,1,1,1,4]; c2=[1,1,1,1,1,1,1,2,1,1,3,1,2,3,3,2,1,1,1,1,1,3,1,2,1,1,1,1,3,3,1,2,1,1,1,3,1, 3,1,2,3,1,3,1,1,3,1,2,3,3,1,3,3,4,1,4,1,3,1,1,1,3,1,2,1,1,4,1,3,3,1,2,3,1, 1,3,1,3,1,2,1,1,1,1,1,3,3,4,1,3,1,1,1,3,1,4,1,1,3,3,1,3,3,2,3,3,1,4,1,3,1, 2,1,1,3,1,1,3,3,4,4,3,3,1,1,3,3,4,1,1,1,1,1,3,1,2,3,1,3,1,3,3,1,6,1,1,1,1, 1,3,1,4,3,1,1,1,3,3,1,2,1,4,3,1,3,3,3,2,3,1,2,1,1,3,4,2,1,3,1,1,1,3,3,6,1, 1,2,1,3,3,1,2,1,1,1,1,1,3,1,6];} a(n) = if(n%4>2,c2[n\2]/sqrt(3),if(n%4>1,0,if(n%4>0,c1[n\2+1],c1[n/2]+c2[n/2]/sqrt(3)))); coefgrow(n) = n^(1/3); \\ approx. growth of the coefficients in general initLdata("a(k)"); \\ Initalized L-series with coefficients a(k) \\ actually uses cflength()=352 coeffs print("EXAMPLE: L(s)=Shintani's zeta function"); print(" with ",default(realprecision)," digits precision"); print("Verifying functional equation. Error: ",errprint(checkfeq())); print("L(2) = ",L(2)); print(" (check) = ",L(2,1.1)); print("L(3) = ",L(3)); print(" (check) = ",L(3,1.1));