Hello,
I would like to know whether there are known examples of primitive functions of the Selberg class having at least one non simple non trivial zero or not. Thank you in advance.
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asked Apr 26, 2011 at 21:05
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2$\begingroup$ As in the examples given below, this should only ever happen at $s=1/2$. $\endgroup$– David HansenCommented Apr 27, 2011 at 3:40
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$\begingroup$ Do you mean that whatever the primitive function of the Selberg class, $s=1/2$ is the only possible non simple non trivial zero? $\endgroup$– Sylvain JULIENCommented Apr 29, 2011 at 0:17
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$\begingroup$ By the way, I would like to know whether the following statement is true or not: "for all positive integer $d$, there is a primitive function of the Selberg class of degree $d$ all the non trivial zeros of which are simple". Thank you in advance. $\endgroup$– Sylvain JULIENCommented May 5, 2011 at 23:53
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2 Answers
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There is an example given in this paper:
J. P. Buhler, B. H. Gross, and D. B. Zagier, "On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3," Mathematics of Computation 44 (1985), no. 170, pp. 473-481.
answered Apr 26, 2011 at 22:02
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Any $L$-function of a (modular) elliptic curve of analytic rank $\geq2$, such as Cremona's curve '389a', will have a non-trivial double zero at $s=1$. More generally, the $L$-function of any irreducible cuspidal automorphic representation of $GL(2,\mathbf{Q})$ for which the Ramanajuan conjecture holds (such as holomorphic cusp forms) is a primitive element of the Selberg class (a result of Ram Murty's MR1354178).
answered Apr 27, 2011 at 1:31