# lfunc_search downloaded from the LMFDB on 26 August 2026. # Search link: https://www.lmfdb.org/L/2/3040/760.499/c0-0 # Query "{'degree': 2, 'conductor': 3040, 'spectral_label': 'c0-0'}" returned 56 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. "2-3040-3040.1709-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false -0.390625 0 0.55740925441364921504112373537 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/2"] "2-3040-3040.1709-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false -0.14062500000000003 0 0.807037847904901476122377726525 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/3"] "2-3040-3040.1709-c0-0-2" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false 0.109375 0 1.10350572633682419028023088950 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/7"] "2-3040-3040.1709-c0-0-3" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false -0.265625 0 1.16390363369579485743407653631 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/8"] "2-3040-3040.1709-c0-0-4" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false 0.23437500000000003 0 1.37970672401797491502599746941 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/1"] "2-3040-3040.1709-c0-0-5" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false -0.015625 0 1.43943600640463496749605405665 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/4"] "2-3040-3040.1709-c0-0-6" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false 0.48437500000000006 0 1.85199848089373443947763673347 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/5"] "2-3040-3040.1709-c0-0-7" 1.2317295313518821 1.5171576384043273 2 3040 "3040.1709" [] [[0.0, 0.0]] 0 true true false false 0.359375 0 2.18758417341662703245753074513 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/1709/6"] "2-3040-3040.189-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false 0.42187500000000006 0 0.27061829123969998515706105178 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/5"] "2-3040-3040.189-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false 0.296875 0 0.67374929746799903186861107987 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/2"] "2-3040-3040.189-c0-0-2" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false -0.328125 0 0.72568875372065699814535023896 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/1"] "2-3040-3040.189-c0-0-3" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false -0.078125 0 1.09571823621088083909084422720 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/4"] "2-3040-3040.189-c0-0-4" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false 0.04687500000000001 0 1.38696588056189908065492800061 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/3"] "2-3040-3040.189-c0-0-5" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false 0.17187500000000003 0 1.49629673416958868005094115038 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/8"] "2-3040-3040.189-c0-0-6" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false -0.20312500000000003 0 1.53583559819028174624479916959 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/7"] "2-3040-3040.189-c0-0-7" 1.2317295313518821 1.5171576384043273 2 3040 "3040.189" [] [[0.0, 0.0]] 0 true true false false -0.453125 0 1.88576112041199998146531585560 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/189/6"] "2-3040-3040.2469-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false -0.359375 0 0.39428597000187573472290969798 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/6"] "2-3040-3040.2469-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false 0.015625 0 0.44407513299358183793494515776 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/4"] "2-3040-3040.2469-c0-0-2" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false 0.14062500000000003 0 0.78328784429321338777483151989 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/3"] "2-3040-3040.2469-c0-0-3" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false -0.48437500000000006 0 1.00961141699652097689882050759 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/5"] "2-3040-3040.2469-c0-0-4" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false -0.23437500000000003 0 1.15999895981725472859911522484 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/1"] "2-3040-3040.2469-c0-0-5" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false -0.109375 0 1.29713672214449542396498264868 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/7"] "2-3040-3040.2469-c0-0-6" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false 0.390625 0 1.38411054141067054627073567722 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/2"] "2-3040-3040.2469-c0-0-7" 1.2317295313518821 1.5171576384043273 2 3040 "3040.2469" [] [[0.0, 0.0]] 0 true true false false 0.265625 0 1.83127295650742479402453418029 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/2469/8"] "2-3040-3040.949-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false -0.296875 0 0.42672120605197690185392452213 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/2"] "2-3040-3040.949-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false 0.453125 0 0.74025334897137071275010847702 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/6"] "2-3040-3040.949-c0-0-2" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false -0.04687500000000001 0 0.75494897701586701171342596483 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/3"] "2-3040-3040.949-c0-0-3" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false -0.17187500000000003 0 1.03005769980918105566605803683 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/8"] "2-3040-3040.949-c0-0-4" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false -0.42187500000000006 0 1.58992096609834232670325275296 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/5"] "2-3040-3040.949-c0-0-5" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false 0.328125 0 1.77073380192371602501402105405 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/1"] "2-3040-3040.949-c0-0-6" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false 0.078125 0 1.91233316741232739971932007705 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/4"] "2-3040-3040.949-c0-0-7" 1.2317295313518821 1.5171576384043273 2 3040 "3040.949" [] [[0.0, 0.0]] 0 true true false false 0.20312500000000003 0 2.02760673763427013323404565610 ["ModularForm/GL2/Q/holomorphic/3040/1/cn/a/949/7"] "2-3040-760.139-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.139" [] [[0.0, 0.0]] 0 true true false false -0.16425500441609855 0 0.68720436466120847680485713214 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/a/2799/1"] "2-3040-760.139-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.139" [] [[0.0, 0.0]] 0 true true false false -0.16425500441609855 0 1.03188143779678336927879676959 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/b/2799/1"] "2-3040-760.189-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false -0.06250000000000001 0 0.59951078393728761524774019815 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/2"] "2-3040-760.189-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false 0.4375 0 0.60574365584322645523386566179 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/8"] "2-3040-760.189-c0-0-2" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false -0.3125 0 0.797008298460014724923355452999 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/6"] "2-3040-760.189-c0-0-3" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false -0.1875 0 0.835846983054331114201909965002 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/5"] "2-3040-760.189-c0-0-4" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false 0.1875 0 1.45958532729500536077602201562 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/4"] "2-3040-760.189-c0-0-5" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false 0.06250000000000001 0 1.52151892859822539392256736196 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/7"] "2-3040-760.189-c0-0-6" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false 0.3125 0 1.64141666770680814925516345457 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/3"] "2-3040-760.189-c0-0-7" 1.2317295313518821 1.5171576384043273 2 3040 "760.189" [] [[0.0, 0.0]] 0 true true false false -0.4375 0 2.23193219135662575691594295956 ["ModularForm/GL2/Q/holomorphic/3040/1/b/a/1329/1"] "2-3040-760.339-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.339" [] [[0.0, 0.0]] 0 true true false false 0.16425500441609855 0 1.56651883601562805623368028268 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/a/719/1"] "2-3040-760.339-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.339" [] [[0.0, 0.0]] 0 true true false false 0.16425500441609855 0 1.58810479876707496338393692004 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/b/719/1"] "2-3040-760.499-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.499" [] [[0.0, 0.0]] 0 true true false false -0.25620274721236413 0 0.25425026458404651203876292289 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/a/879/1"] "2-3040-760.499-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.499" [] [[0.0, 0.0]] 0 true true false false -0.25620274721236413 0 0.77383324126801386941563652194 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/b/879/1"] "2-3040-760.539-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.539" [] [[0.0, 0.0]] 0 true true false false -0.13279002263552828 0 0.74695730338733601657553426168 ["ModularForm/GL2/Q/holomorphic/3040/1/cc/a/1679/1", "ArtinRepresentation/2.3040.12t18.b.a"] "2-3040-760.539-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.539" [] [[0.0, 0.0]] 0 true true false false -0.13279002263552828 0 0.937359203617292703491779511416 ["ModularForm/GL2/Q/holomorphic/3040/1/cc/b/1679/1", "ArtinRepresentation/2.3040.12t18.c.a"] "2-3040-760.579-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.579" [] [[0.0, 0.0]] 0 true true false false 0.1085955679015395 0 1.03919453577127541743804489423 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/b/2479/1"] "2-3040-760.579-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.579" [] [[0.0, 0.0]] 0 true true false false 0.1085955679015395 0 1.75863597277372425756512297595 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/a/2479/1"] "2-3040-760.619-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.619" [] [[0.0, 0.0]] 0 true true false false 0.13279002263552828 0 1.29828814189414780789059437273 ["ModularForm/GL2/Q/holomorphic/3040/1/cc/b/239/1", "ArtinRepresentation/2.3040.12t18.c.b"] "2-3040-760.619-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.619" [] [[0.0, 0.0]] 0 true true false false 0.13279002263552828 0 1.48731107266169626390219365833 ["ModularForm/GL2/Q/holomorphic/3040/1/cc/a/239/1", "ArtinRepresentation/2.3040.12t18.b.b"] "2-3040-760.739-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.739" [] [[0.0, 0.0]] 0 true true false false -0.1085955679015395 0 0.64067467067672329575512272227 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/b/2639/1"] "2-3040-760.739-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.739" [] [[0.0, 0.0]] 0 true true false false -0.1085955679015395 0 1.17935930132678340786929261471 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/a/2639/1"] "2-3040-760.99-c0-0-0" 1.2317295313518821 1.5171576384043273 2 3040 "760.99" [] [[0.0, 0.0]] 0 true true false false 0.25620274721236413 0 1.23882706076437555797219738064 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/a/1999/1"] "2-3040-760.99-c0-0-1" 1.2317295313518821 1.5171576384043273 2 3040 "760.99" [] [[0.0, 0.0]] 0 true true false false 0.25620274721236413 0 1.79108073477511873239768111000 ["ModularForm/GL2/Q/holomorphic/3040/1/dv/b/1999/1"] # 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).