/*** EXAMPLE: Modular form Delta ***/
/*** (illustration of unusual coefficient growth) ***/
/*** ***/
/*** v1.2, July 2013, questions to tim.dokchitser@bristol.ac.uk ***/
/*** type \rex-delta or read("ex-delta") at pari prompt to run this ***/
read("computel"); \\ read the ComputeL package
\\ and set the default values
default(realprecision,40); \\ set working precision; used throughout
\\ * Coefficients are given by Ramanujan's tau function
\\ * Re-define the default bound on the coefficients:
\\ Deligne's estimate on tau(n) is better than the default
\\ coefgrow(n)=(4n)^(11/2) (by a factor 1024),
\\ so re-defining coefgrow() improves efficiency (slightly faster)
coefgrow(n) = 2*n^(11/2);
tau(n) = (5*sigma(n,3)+7*sigma(n,5))*n/12\
-35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5));
\\ initialize L-function parameters
conductor = 1; \\ exponential factor
gammaV = [0,1]; \\ list of gamma-factors
weight = 12; \\ L(s)=sgn*L(weight-s)
sgn = 1; \\ sign in the functional equation
initLdata("tau(k)"); \\ L-series coefficients a(k)
print("EXAMPLE: L-function associated to the modular form Delta of weight 12");
print(" coefficients = Ramanujan's tau function");
print(" with ",default(realprecision)," digits precision");
print("Verifying functional equation. Error: ",errprint(checkfeq()));
print("L(1) = ",lval = L(1));
print(" (check) = ",lval2 = L(1,1.1)," (err=",errprint(lval-lval2),")");