Rational L-functions in LMFDB with Root Analytic Conductor Less Than 4

The dataset includes all the rational L-functions in LMFDB with root analytic conductor less than 4 present on January 30, 2025.

The dataset is presented as text file with an header and the following columns separated by a column :.

• label: the L-function label
• primitive: a boolean indentifying if the L-functions is primitive, i.e., not a product of other not-necessarily-rational L-functions.
• conductor: the conductor of the L-function.
• central_character: the label N.q of the central character, which identifies a Dirichlet character.
• motivic_weight: the motivic weight of the L-function.
• degree: the degree of the L-function (the degree of its Euler factors at primes away from the conductor).
• order_of_vanishing: order of vanishing at its central point, also known as analytic rank
• z1: the lowest zero on the critical line
• root_angle: the root angle of the L-function
• root_analytic_conductor: is the root analytic conductor of the L-function.
• instance_types: sequence representing the types of the several known underlying objects of an L-function that are in the LMFDB.
• ap: is a list of the first Dirichlet coefficients (arithmetically normalized) at primes less than 1000

With the exception of the column ap, these are all columns already present in the LMFDB.

The column ap was computed aposteri. The instance_types value for the L-functions with label 4-1879e2-1.1-c1e2-0-0 and 4-643e2-1.1-c1e2-0-0 was updated to include BMF, as theoretically we know this to also be the case.