Completeness of L-function Data (reviewed)

The database currently contains more than $20$ million L-functions. Most of these are stored in the database along with related information such as zeros and analytic rank that have been precomputed using explicit precision bounds (see the “reliability” link on each L-function's home page for further details). A small number of dynamically computed L-functions are also available, as noted in the summary below. The pages for dynamically computed L-functions may be missing information that cannot be computed quickly, and the data displayed on them may be less accurate than the displayed precision suggests.

Below is a summary of available L-functions by degree. Note that the counts are not mutually exclusive, as different types of mathematical objects may have the same L-function. For precomputed L-functions, objects in the database that are believed to give rise to this L-function are listed in the “origins” box on the L-functions home page.

Degree 1

• The Riemann zeta-function $\zeta(s)$.
• $7\,655\,469$ Dirichlet L-functions associated to primitive Dirichlet characters, including all $1\,448\,484$ of conductor $N\le 2800$ (this includes the Riemann zeta-function).

Degree 2

• $14\,398\,359$ L-functions of classical modular forms (more precisely, embedded newforms); this includes the L-functions of all newforms in the database that have weight $k\le 200$ (motivic weight $w=k-1\le 199$), including $152\,268$ Artin L-functions corresponding to weight $1$ newforms ($6470$ of the corresponding Artin representations are in the database).
• $1\,741\,002$ L-functions of elliptic curves over $\Q$ corresponding to the $1\,741\,002$ isogeny classes of conductor $N\le 400\,000$; of these, $38\,042$ have conductor $N\le 10\,000$ and correspond to L-functions of newforms in the database (those of weight $2$, trivial character, and dimension $1$).
• $15\,659$ dynamically computed L-functions of $\GL(2)$ Maass forms.
• $4296$ dynamically computed Dedekind zeta-functions of the quadratic fields with absolute discriminant $|D|\le 7071$ are available.
• $55\,810$ Artin L-functions arising from $2$-dimensional Artin representations of conductor $N\le 27\,000\,000$, of which all but $6470$ are dynamically computed.

Degree 3

• $1552$ L-functions of $\GL(3)$ Maass forms.
• $51$ dynamically computed Dedekind zeta functions of the cubic fields with absolute discriminant $|D|\le 368$.
• $7925$ dynamically computed Artin L-functions arising from $3$-dimensional Artin representations of conductor $N\le 90\,000$ are available.

Degree 4

• $41\,653$ imprimitive rational L-functions corresponding to (Galois orbits of) newforms of dimension $2$.
• $253\,190$ L-functions of elliptic curve isogeny classes over quadratic fields (all but $9$ isogeny classes of elliptic curves over quadratic fields in the database).
• $66\,684$ L-functions of Hilbert modular forms of dimension $1$ over real quadratic fields (each is also the L-function of an isogeny class of elliptic curves over a quadratic field).
• $2913$ dynamically computed L-functions of Hilbert modular forms of dimension $d>1$ over real quadratic fields.
• $185\,541$ L-functions of Bianchi newforms of dimension $1$ over imaginary quadratic fields (all but the $40$ that do not correspond to elliptic curves).
• $65\,534$ L-functions of isogeny classes of genus $2$ curves over $\Q$ (some of which also arise as L-functions of classical/Hilbert/Bianchi newforms, elliptic curves over quadratic fields, or products of L-functions of elliptic curves over $\Q$).
• $35$ dynamically computed Artin L-functions arising from $4$-dimensional Artin representations of conductor $N\le 5196$ are available.

** Degrees 5 to 40 **

• $115\,645$ imprimitive rational L-functions corresponding to newforms of dimensions $3$ to $20$ and weight $k\le 200$ (there are at least $2000$ in each even degree from $6$ to $40$).