Intrinsic Criteria for Expected L-Function Properties

Definitions and mathematical statement

Write $\mathcal M$ for the intended class of arithmetic or geometric objects and $L(M,s)$ for the intended $L$-function associated with an object $M\in\mathcal M$. These are placeholders: the supplied record specifies neither the class $\mathcal M$ nor the construction or initial domain of $L(M,s)$.

Research question. Determine intrinsic conditions on $M$ that ensure, for every object satisfying those conditions, that its associated $L$-function has the expected local-to-global properties:

1. an Euler product, meaning a factorization into the appropriate local factors;
2. an archimedean factor, supplying the appropriate contribution at infinity;
3. meromorphic continuation;
4. a functional equation.

The original statement calls these "local-to-global interpolation properties," but supplies no definition of interpolation in this setting.

Unresolved formulation. The record does not specify a base field, the local indexing set or factors, a convergence domain for the Euler product, the archimedean normalization, the domain of meromorphic continuation, or the form and normalization of the functional equation. It also does not define which properties count as intrinsic. Thus the list describes the requested objectives, not a fully specified analytic predicate or an asserted theorem. A precise quantified proposition requires these choices and a specified class of objects; none is selected here.

Short English statement

The question asks which intrinsic features of an arithmetic or geometric object force its associated $L$-function to have an Euler product, an archimedean factor, meromorphic continuation, and a functional equation. The supplied formulation leaves the objects, the associated functions, and the precise analytic requirements unspecified, so it remains a foundational research question requiring further formulation.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30002407$, source problem code catalogue entry, titled "Intrinsic Criteria for Expected L-Function Properties," in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Noncommutative Geometry (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/45, pp. 2553--2629. https://doi.org/10.4171/owr/2013/45. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. It records proposed year 2013 and identifies no individual proposer. The relationship between the citation's 2014 label and the recorded 2013 year is unresolved here.
• Supplied literature source, labeled "Oberwolfach Report": https://ems.press/content/serial-article-files/46476.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes it as a broad foundational question rather than a precise proposition, with no supplied criterion established or refuted. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The source requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was unchanged while notation, typography, and wording were normalized into a self-contained publication form. This is a supplied editorial qualification, not independent mathematical verification of this draft.