I would like to know whether Langlands' functoriality conjecture implies that the Selberg class coincides with the class of automorphic L-functions and, if so, whether this class is closed under Rankin-Selberg convolution or not.

Many thanks in advance.


The Langlands functoriality conjecture implies that automorphic $L$-functions belong to the Selberg class, but not the other way (i.e. the other direction is not known to follow from this conjecture). Regarding to Rankin-Selberg convolutions, I don't think that this operation has been defined precisely for the Selberg class. There is a subtlety at the ramified primes: one usually refers to the algebraic classification of the underlying local representations, which is itself part of the Langlands program.

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  • $\begingroup$ If I'm not mistaken, this operation has been defined for the Selberg class in Murty, M. Ram and Zaharescu, Alexandru (2002), ”Explicit for- mulas for the pair correlation of zeros of functions in the Selberg class”, Forum Math. 14 (2002), no. 1, pp. 65-83. $\endgroup$ – Sylvain JULIEN Jan 31 '16 at 18:24
  • $\begingroup$ may I send you a copy of the latest version of an article of mine dealing with such issues? You can join me at sylvainjul'at'gmail'dot'com. $\endgroup$ – Sylvain JULIEN Jan 31 '16 at 18:32
  • $\begingroup$ @SylvainJULIEN: I don't have any free time to join your project, sorry about this. If your paper gets published, I will be happy to look at it. $\endgroup$ – GH from MO Jan 31 '16 at 18:52
  • $\begingroup$ You can read it there: les-mathematiques.net/phorum/read.php?43,1210447 $\endgroup$ – Sylvain JULIEN Feb 2 '16 at 14:05

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