# lfunc_search downloaded from the LMFDB on 17 April 2026. # Search link: https://www.lmfdb.org/L/rational/4/120^2 # Query "{'degree': 4, 'conductor': 14400, 'rational': True}" returned 44 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. "4-120e2-1.1-c0e2-0-0" 0.24471984291421955 0.003586548770415056 4 14400 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 2.15313362465143136876203290842 ["ModularForm/GL2/Q/holomorphic/120/1/i/a", "ArtinRepresentation/2.120.8t11.c"] "4-120e2-1.1-c1e2-0-0" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.70139200864348904989459807621135 ["ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.3-c", "EllipticCurve/2.2.24.1/25.3/c", "ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.2-c", "EllipticCurve/2.2.24.1/25.2/c"] "4-120e2-1.1-c1e2-0-1" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.870154925974925884967934945032102 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/225.2/a", "EllipticCurve/2.0.8.1/225.2/a"] "4-120e2-1.1-c1e2-0-10" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 1.63394710947634814954509320812148 ["ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-576.1-c", "EllipticCurve/2.2.5.1/576.1/b", "EllipticCurve/2.2.5.1/576.1/c", "ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-576.1-b"] "4-120e2-1.1-c1e2-0-11" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.63558305615 ["Genus2Curve/Q/14400/e"] "4-120e2-1.1-c1e2-0-12" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.75820205156 ["Genus2Curve/Q/14400/a"] "4-120e2-1.1-c1e2-0-13" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 1.76844443893169993470264607728055 ["ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.2-d", "EllipticCurve/2.2.24.1/25.2/d", "ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.3-d", "EllipticCurve/2.2.24.1/25.3/d"] "4-120e2-1.1-c1e2-0-14" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.81793015252092636076156145980324 ["EllipticCurve/2.2.12.1/100.1/a", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-100.1-a"] "4-120e2-1.1-c1e2-0-15" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.88409811084955135522838104472 ["EllipticCurve/2.2.5.1/576.1/a", "ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-576.1-a", "ModularForm/GL2/Q/holomorphic/120/2/f/a"] "4-120e2-1.1-c1e2-0-16" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 1.92321555608603079637936047187519 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/225.1/a", "EllipticCurve/2.0.8.1/225.1/a", "EllipticCurve/2.0.8.1/225.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/225.3/a"] "4-120e2-1.1-c1e2-0-17" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 2.06437956968 ["Genus2Curve/Q/14400/c"] "4-120e2-1.1-c1e2-0-18" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 2.08110417889891008752239981227984 ["EllipticCurve/2.2.24.1/25.2/a", "ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.3-a", "ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.2-a", "EllipticCurve/2.2.24.1/25.3/a"] "4-120e2-1.1-c1e2-0-19" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 2.11486296578438549266961855723 ["ModularForm/GL2/TotallyReal/2.2.8.1/holomorphic/2.2.8.1-225.1-a", "EllipticCurve/2.2.8.1/225.1/a", "ModularForm/GL2/Q/holomorphic/120/2/k/a"] "4-120e2-1.1-c1e2-0-2" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.03399709962 ["Genus2Curve/Q/14400/d"] "4-120e2-1.1-c1e2-0-20" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 2.22546285886 ["Genus2Curve/Q/14400/f"] "4-120e2-1.1-c1e2-0-21" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 2.23236505279691350596522634765065 ["ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.3-b", "ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.2-b", "EllipticCurve/2.2.24.1/25.2/b", "EllipticCurve/2.2.24.1/25.3/b"] "4-120e2-1.1-c1e2-0-22" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 2.61772465980045468774058435808703 ["EllipticCurve/2.0.3.1/1600.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/1600.1/a"] "4-120e2-1.1-c1e2-0-3" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.14288269174 ["Genus2Curve/Q/14400/b"] "4-120e2-1.1-c1e2-0-4" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.16915866227454488376692012453777 ["ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-100.1-b", "EllipticCurve/2.2.12.1/100.1/b"] "4-120e2-1.1-c1e2-0-5" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.18815390030287753587945796282149 ["ModularForm/GL2/TotallyReal/2.2.8.1/holomorphic/2.2.8.1-225.1-b", "EllipticCurve/2.2.8.1/225.1/b"] "4-120e2-1.1-c1e2-0-6" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 1.33701408024699827847316895761617 ["EllipticCurve/2.0.4.1/900.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/900.1/a", "EllipticCurve/2.0.4.1/900.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/900.3/a"] "4-120e2-1.1-c1e2-0-7" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.38738099288330045270314903054332 ["ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-576.1-e", "EllipticCurve/2.2.5.1/576.1/e"] "4-120e2-1.1-c1e2-0-8" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 1.47164565177321586546875829968365 ["ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-576.1-d", "ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-576.1-f", "EllipticCurve/2.2.5.1/576.1/f", "EllipticCurve/2.2.5.1/576.1/d"] "4-120e2-1.1-c1e2-0-9" 0.9788793716568784 0.9181564852262551 4 14400 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 1.54977855646621818940608154138434 ["ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.2-e", "ModularForm/GL2/TotallyReal/2.2.24.1/holomorphic/2.2.24.1-25.3-e", "EllipticCurve/2.2.24.1/25.2/e", "EllipticCurve/2.2.24.1/25.3/e"] "4-120e2-1.1-c3e2-0-0" 2.6608700082283008 50.129645534287455 4 14400 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.68348148316533899459380789864 ["ModularForm/GL2/Q/holomorphic/120/4/f/b"] "4-120e2-1.1-c3e2-0-1" 2.6608700082283008 50.129645534287455 4 14400 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.74491811267364080978435723587 ["ModularForm/GL2/Q/holomorphic/120/4/f/c"] "4-120e2-1.1-c3e2-0-2" 2.6608700082283008 50.129645534287455 4 14400 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.796257521550782573282625880415 ["ModularForm/GL2/Q/holomorphic/120/4/k/a"] "4-120e2-1.1-c3e2-0-3" 2.6608700082283008 50.129645534287455 4 14400 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 2.23215940033862635561626329548 ["ModularForm/GL2/Q/holomorphic/120/4/f/a"] "4-120e2-1.1-c5e2-0-0" 4.387032981134024 370.41076307235824 4 14400 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 1.06208173210261488891787289601 ["ModularForm/GL2/Q/holomorphic/120/6/a/g"] "4-120e2-1.1-c5e2-0-1" 4.387032981134024 370.41076307235824 4 14400 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 1.16464903857829264563311272051 ["ModularForm/GL2/Q/holomorphic/120/6/a/h"] "4-120e2-1.1-c7e2-0-0" 6.122597737733113 1405.215419712703 4 14400 "1.1" [] [[3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.43498753171079083137793286662 ["ModularForm/GL2/Q/holomorphic/120/8/a/c"] "4-120e2-1.1-c7e2-0-1" 6.122597737733113 1405.215419712703 4 14400 "1.1" [] [[3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.68172775311786479338761953644 ["ModularForm/GL2/Q/holomorphic/120/8/a/f"] "4-120e2-1.1-c7e2-0-2" 6.122597737733113 1405.215419712703 4 14400 "1.1" [] [[3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.74994107117797326048101225354 ["ModularForm/GL2/Q/holomorphic/120/8/a/g"] "4-120e2-1.1-c7e2-0-3" 6.122597737733113 1405.215419712703 4 14400 "1.1" [] [[3.5, 0.0], [3.5, 0.0]] 7 false 1 0 1.01857119446272626512192141666 ["ModularForm/GL2/Q/holomorphic/120/8/a/h"] "4-120e2-1.1-c7e2-0-4" 6.122597737733113 1405.215419712703 4 14400 "1.1" [] [[3.5, 0.0], [3.5, 0.0]] 7 false 1 2 1.47111502929956369854570006369 ["ModularForm/GL2/Q/holomorphic/120/8/a/d"] "4-120e2-1.1-c7e2-0-5" 6.122597737733113 1405.215419712703 4 14400 "1.1" [] [[3.5, 0.0], [3.5, 0.0]] 7 false 1 2 1.55429825812818530855541502588 ["ModularForm/GL2/Q/holomorphic/120/8/a/e"] "4-120e2-1.1-c9e2-0-0" 7.861571111404183 3819.7715404754904 4 14400 "1.1" [] [[4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.21630376251273011202393870095 ["ModularForm/GL2/Q/holomorphic/120/10/a/b"] "4-120e2-1.1-c9e2-0-1" 7.861571111404183 3819.7715404754904 4 14400 "1.1" [] [[4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.63855920570838971762807598078 ["ModularForm/GL2/Q/holomorphic/120/10/a/c"] "4-120e2-1.1-c9e2-0-2" 7.861571111404183 3819.7715404754904 4 14400 "1.1" [] [[4.5, 0.0], [4.5, 0.0]] 9 false 1 2 1.25617178876299316835605591863 ["ModularForm/GL2/Q/holomorphic/120/10/a/a"] "4-120e2-1.1-c9e2-0-3" 7.861571111404183 3819.7715404754904 4 14400 "1.1" [] [[4.5, 0.0], [4.5, 0.0]] 9 false 1 2 1.42126167702296663276675597646 ["ModularForm/GL2/Q/holomorphic/120/10/a/d"] "4-120e2-1.1-c9e2-0-4" 7.861571111404183 3819.7715404754904 4 14400 "1.1" [] [[4.5, 0.0], [4.5, 0.0]] 9 false 1 2 1.55792782556272031589823554164 ["ModularForm/GL2/Q/holomorphic/120/10/a/e"] "4-120e2-1.1-c9e2-0-5" 7.861571111404183 3819.7715404754904 4 14400 "1.1" [] [[4.5, 0.0], [4.5, 0.0]] 9 false 1 2 1.64299354930208739596711431095 ["ModularForm/GL2/Q/holomorphic/120/10/a/f"] "4-120e2-1.1-c11e2-0-0" 9.60214463894138 8501.057900818923 4 14400 "1.1" [] [[5.5, 0.0], [5.5, 0.0]] 11 false 1 2 1.44083864792468105487828886890 ["ModularForm/GL2/Q/holomorphic/120/12/a/a"] "4-120e2-1.1-c11e2-0-1" 9.60214463894138 8501.057900818923 4 14400 "1.1" [] [[5.5, 0.0], [5.5, 0.0]] 11 false 1 2 1.49248588297175867632180470695 ["ModularForm/GL2/Q/holomorphic/120/12/a/b"] # 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).