# lfunc_search downloaded from the LMFDB on 11 April 2026. # Search link: https://www.lmfdb.org/L/rational/4/578^2 # Query "{'degree': 4, 'conductor': 334084, 'rational': True}" returned 23 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, $\epsilon$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "4-578e2-1.1-c1e2-0-0" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.57562187814673446242727734974 ["ModularForm/GL2/Q/holomorphic/578/2/c/c"] "4-578e2-1.1-c1e2-0-1" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.874603455084946305049555057876 ["ModularForm/GL2/Q/holomorphic/578/2/b/b"] "4-578e2-1.1-c1e2-0-2" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.903121005584487045429289479848 ["ModularForm/GL2/Q/holomorphic/578/2/b/a"] "4-578e2-1.1-c1e2-0-3" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.16446472124218801653543660700 ["ModularForm/GL2/Q/holomorphic/578/2/a/c"] "4-578e2-1.1-c1e2-0-4" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.16659961133562296556166997994 ["ModularForm/GL2/Q/holomorphic/578/2/c/a"] "4-578e2-1.1-c1e2-0-5" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.19639459271655295401010678152 ["ModularForm/GL2/Q/holomorphic/578/2/b/c"] "4-578e2-1.1-c1e2-0-6" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.21675379765085187782761711890 ["ModularForm/GL2/Q/holomorphic/578/2/c/b"] "4-578e2-1.1-c1e2-0-7" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.24415469407153151063526615244 ["ModularForm/GL2/Q/holomorphic/578/2/a/d"] "4-578e2-1.1-c1e2-0-8" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.34684557834792041633322747134 ["ModularForm/GL2/Q/holomorphic/578/2/c/d"] "4-578e2-1.1-c1e2-0-9" 2.14833731914474 21.301485500717238 4 334084 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.94947439300087123984513351010 ["ModularForm/GL2/Q/holomorphic/578/2/a/b"] "4-578e2-1.1-c3e2-0-0" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.24738779603131931064756858526 ["ModularForm/GL2/Q/holomorphic/578/4/a/f"] "4-578e2-1.1-c3e2-0-1" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.44627555612345498115236395832 ["ModularForm/GL2/Q/holomorphic/578/4/a/g"] "4-578e2-1.1-c3e2-0-2" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.52999250443950226087006379449 ["ModularForm/GL2/Q/holomorphic/578/4/b/a"] "4-578e2-1.1-c3e2-0-3" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.66138972632745195608683803445 ["ModularForm/GL2/Q/holomorphic/578/4/b/b"] "4-578e2-1.1-c3e2-0-4" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 0.934557165664286796758956445162 ["ModularForm/GL2/Q/holomorphic/578/4/a/e"] "4-578e2-1.1-c3e2-0-5" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.992360557399935760049731333673 ["ModularForm/GL2/Q/holomorphic/578/4/b/c"] "4-578e2-1.1-c3e2-0-6" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 1.18788204239851333407030395366 ["ModularForm/GL2/Q/holomorphic/578/4/a/i"] "4-578e2-1.1-c3e2-0-7" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.56939575787786529930346526750 ["ModularForm/GL2/Q/holomorphic/578/4/a/k"] "4-578e2-1.1-c3e2-0-8" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.80939474882029838927334456772 ["ModularForm/GL2/Q/holomorphic/578/4/a/h"] "4-578e2-1.1-c3e2-0-9" 5.839786296031567 1163.0217012970063 4 334084 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.82011368831826048008149456060 ["ModularForm/GL2/Q/holomorphic/578/4/a/j"] "4-578e2-1.1-c5e2-0-0" 9.62817988261036 8593.632595157342 4 334084 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 0.878856773889731073324969919943 ["ModularForm/GL2/Q/holomorphic/578/6/a/c"] "4-578e2-1.1-c5e2-0-1" 9.62817988261036 8593.632595157342 4 334084 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 1.24073265658407007821882372597 ["ModularForm/GL2/Q/holomorphic/578/6/a/b"] "4-578e2-1.1-c7e2-0-0" 13.437207475134946 32601.388074951294 4 334084 "1.1" [] [[3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.41613298387882610980491886330 ["ModularForm/GL2/Q/holomorphic/578/8/a/e"] # 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 $1$. #$\epsilon$ (root_number) -- # The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$. The sign appears as the 4th entry in the quadruple # known as the Selberg data. #$r$ (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).