Usually the model and relevant product or service reference are also needed. A serial number identifies an individual appliance within a manufacturer's system, not necessarily a publicly searchable part family.
What the observation tells you
Sending only a long number can create confusion about whether it is a serial, model or production code. A shelf also has a particular position and may differ from another shelf in the same fridge. Dimensions are useful supporting information, not complete compatibility evidence.
A useful next step
Provide the whole rating label and specify the shelf location. Keep damaged parts unloaded while suitability is checked. Do not trim or adapt a nearly matching shelf.
Manuals and supporting guidance
Model-specific instructions take priority. These references support the practical principles discussed here; they do not diagnose your appliance.