# lfunc_search downloaded from the LMFDB on 23 May 2024.
# Search link: https://www.lmfdb.org/L/1/5%5E2/25.13/r1-0
# Query "{'degree': 1, 'conductor': 25, 'spectral_label': 'r1-0'}" returned 8 lfunc_searchs, sorted by root analytic conductor.
# Each entry in the following data list has the form:
# [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin]
# For more details, see the definitions at the bottom of the file.
"1-5e2-25.12-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.12" [[1, 0.0]] [] 0 true true false false 0.28 0 2.53697086842 ["Character/Dirichlet/25/12"]
"1-5e2-25.13-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.13" [[1, 0.0]] [] 0 true true false false -0.22 0 1.58220676001 ["Character/Dirichlet/25/13"]
"1-5e2-25.17-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.17" [[1, 0.0]] [] 0 true true false false 0.21000000000000002 0 2.14844988943 ["Character/Dirichlet/25/17"]
"1-5e2-25.2-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.2" [[1, 0.0]] [] 0 true true false false 0.22 0 3.29941230419 ["Character/Dirichlet/25/2"]
"1-5e2-25.22-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.22" [[1, 0.0]] [] 0 true true false false -0.21000000000000002 0 2.00962811208 ["Character/Dirichlet/25/22"]
"1-5e2-25.23-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.23" [[1, 0.0]] [] 0 true true false false -0.28 0 1.08985590213 ["Character/Dirichlet/25/23"]
"1-5e2-25.3-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.3" [[1, 0.0]] [] 0 true true false false -0.21000000000000002 0 0.396445550335 ["Character/Dirichlet/25/3"]
"1-5e2-25.8-r1-0-0" 2.6866231198656254 2.6866231198656254 1 25 "25.8" [[1, 0.0]] [] 0 true true false false 0.21000000000000002 0 3.23341870291 ["Character/Dirichlet/25/8"]
# Label --
# Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where
# * $d$ is the degree of $L$.
# * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions.
# * q.k is the label of the primitive Dirichlet character from which the central character is induced.
# * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation.
# * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label.
#$\alpha$ (root_analytic_conductor) --
# If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields.
#$A$ (analytic_conductor) --
# The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number
# \[
# A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N.
# \]
#$d$ (degree) --
# The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation
# \[
# \Lambda(s) := N^{s/2}
# \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
# \cdot L(s) = \varepsilon \overline{\Lambda}(1-s).
# \]
# The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor.
#$N$ (conductor) --
# The **conductor** of an L-function is the integer $N$ occurring in its functional equation
# \[
# \Lambda(s) := N^{s/2}
# \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
# \cdot L(s) = \varepsilon \overline{\Lambda}(1-s).
# \]
# The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function
# associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$.
# In the literature, the word _level_ is sometimes used instead of _conductor_.
#$\chi$ (central_character) --
# An L-function has an Euler product of the form
# $L(s) = \prod_p L_p(p^{-s})^{-1}$
# where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function.
# Here, $N$ is the conductor of $L$.
#$\mu$ (mus) --
# All known analytic L-functions have a **functional equation** that can be written in the form
# \[
# \Lambda(s) := N^{s/2}
# \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
# \cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
# \]
# where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer
# or half-integer,
# \[
# \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real},
# \]
# and $\varepsilon$ is the sign of the functional equation.
# With those restrictions on the spectral parameters, the
# data in the functional equation is specified uniquely. The integer $d = J + 2 K$
# is the degree of the L-function. The integer $N$ is the conductor (or level)
# of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters
# in the functional equation can be used to make up the 4-tuple called the Selberg data.
# The axioms of the Selberg class are less restrictive than
# given above.
# Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.
# For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$,
# called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.
#$\nu$ (nus) --
# All known analytic L-functions have a **functional equation** that can be written in the form
# \[
# \Lambda(s) := N^{s/2}
# \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
# \cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
# \]
# where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer
# or half-integer,
# \[
# \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real},
# \]
# and $\varepsilon$ is the sign of the functional equation.
# With those restrictions on the spectral parameters, the
# data in the functional equation is specified uniquely. The integer $d = J + 2 K$
# is the degree of the L-function. The integer $N$ is the conductor (or level)
# of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters
# in the functional equation can be used to make up the 4-tuple called the Selberg data.
# The axioms of the Selberg class are less restrictive than
# given above.
# Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.
# For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$,
# called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.
#$w$ (motivic_weight) --
# The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$.
# If the L-function arises from a motive, then the weight of the motive has the
# same parity as the motivic weight of the L-function, but the weight of the motive
# could be larger. This apparent discrepancy comes from the fact that a Tate twist
# increases the weight of the motive. This corresponds to the change of variables
# $s \mapsto s + j$ in the L-function of the motive.
#prim (primitive) --
# An L-function is **primitive** if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$.
#arith (algebraic) --
# An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers.
#$\mathbb{Q}$ (rational) --
# A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes
# \[
# L(s)=\prod_pL_p(p^{-s})^{-1}
# \]
# with $L_p\in \Z[T]$.
#self-dual (self_dual) --
# An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real.
#$\operatorname{Arg}(\epsilon)$ (root_angle) --
# The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$.
#$r$ (order_of_vanishing) --
# The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point.
# When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point).
#First zero (z1) --
# The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$.
# Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization).
# The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number.
#Origin (instance_urls) --
# L-functions arise from many different sources. Already in degree 2 we have examples of
# L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions).
# Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).