5. Other families of $L$functions

Problem 5.1.
[B. Conrey] Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$functions. What new terms contribute to the main term when $k\geq 3$? 
Problem 5.2.
[B. Conrey and H. Iwaniec] Develop a heuristic for moments of $L$functions of automorphic forms.
Remark. [B. Conrey] One of the problems here is that $\sum a_m a_{m+h}=0$, so there are no Type I terms.


Problem 5.3.
[C. TurnageButterbaugh] Develop a heuristic for moments of imprimitive $L$functions, or if your $L$function factors into degree $1$ factors, e.g. $$ \int L(\tfrac{1}{2}+it,\chi_1)\cdots L(\tfrac{1}{2}+it,\chi_k)^2\,dt $$ (or different powers). 
Problem 5.4.
[M. Milinovich] Develop a heuristic for quadratic twists of $L$functions
Remark. [H. Iwaniec] What role does the root number play?


Problem 5.5.
[H. Iwaniec and K. Soundararajan] The mechanism for computing moments in families like $\chi$ (mod $q$), $q\leq Q$, involve different orthogonality relations than $mn=$small. Yet the answers are the same. Why? 
Problem 5.6.
[M. Radziwill] Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH? 
Problem 5.7.
[O. Balkanova] Find an exact formula for $\sum_{f\in H_k}^h L(\frac{1}{2},Sym^2f)^2$.
Cite this as: AimPL: Moments of zeta and correlations of divisor sums, available at http://aimpl.org/zetamoments.