Siegel Modular forms

Consider the spinor zeta-function of a degree 2 weight 2 Siegel modular form. This would be a degree 4 L-function in the Selberg Class terminology. Bocherer has conjectured a special value for the central value of its quadratic twist by a fundamental discriminant $d$. This conjecture implies a discretization of these central values, perhaps by $1/\vert d\vert$. Can random matrix theory be used to predict the frequency of vanishing to order two of the L-functions of these quadratic twists? What are the arithmetic consequences of such vanishing?

It would be great to produce a concrete example here and test it. What is the first level for which a weight 2 Siegel modular form (of degree 2) exists? How do we find the coefficients of its spinor zeta-function. Can we compute the cenral values of the quadratic twists, either directly or by Bocherer's conjecture?

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