show · lfunction.spectral_label all knowls · up · search:

The spectral parameters are encoded as a string "x-y". Here x is the concatenation of expressions of the form r$u$ and c$v$. The first indicates that $\Gamma_{\R}(s + \mu)$ appears in the functional equation with $\Re(\mu) = u$, and the second that $\Gamma_{\C}(s + \nu)$ appears with $2\Re(\nu) = v$. The factor of $2$ is included so that the $u$ and $v$s are integral. In the multiset of $u$ and $v$s, if the multiplicities are all a multiple of an integer $n > 1$ then we use e$n$ to shorten x. For example, r0r0c0c0 is encoded r0c0e2. The $\Gamma_{\R}$ terms and $\Gamma_{\C}$ terms are separately sorted, and we use the identity $\Gamma_{\C}(s) = \Gamma_{\R}(s)\Gamma_{\R}(s + 1)$ as much as possible to combine pairs $\Gamma_{\R}(s + \mu)\Gamma_{\R}(s + \mu + 1)$ into $\Gamma_{\C}(s + \mu)$ in the interest of brevity.

Similarly, y is either 0 (indicating that all $\mu_j$ and $\nu_j$ are real), or the concatenation of expressions of the form p$t$, m$t$ or c$t$ indicating complex numbers with imaginary part either $t$ or $-t$ or both (respectively). These imaginary parts are listed in the same order as the real parts and are sorted by increasing absolute value within parameters with the same real part. We round imaginary parts to two digits after the decimal point. We write p0 when the imaginary part is zero to the known precision, and p0.00 and m0.00 for imaginary parts with an absolute value smaller than $0.5 \cdot 10^{-2}$.

Authors:
Knowl status:
• Review status: beta
• Last edited by David Roe on 2021-01-11 22:08:42
Referred to by:
History: