# lfunc_search downloaded from the LMFDB on 18 April 2026. # Search link: https://www.lmfdb.org/L/2/2852/2852.827/c0-0 # Query "{'degree': 2, 'conductor': 2852, 'spectral_label': 'c0-0'}" returned 98 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. "2-2852-2852.1195-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.1195" [] [[0.0, 0.0]] 0 true true false false 0.046819901161216974 0 0.853932180845454549675650807084 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1195/1"] "2-2852-2852.1195-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.1195" [] [[0.0, 0.0]] 0 true true false false -0.19762454328322748 0 0.949198648004511225084283060740 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1195/3"] "2-2852-2852.1195-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.1195" [] [[0.0, 0.0]] 0 true true false false 0.046819901161216974 0 1.34696316297836030341184166917 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1195/3"] "2-2852-2852.1195-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.1195" [] [[0.0, 0.0]] 0 true true false false 0.13570879005010586 0 1.45325941577402500378897038632 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1195/2"] "2-2852-2852.1195-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.1195" [] [[0.0, 0.0]] 0 true true false false 0.046819901161216974 0 1.46280725321306707241101050389 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1195/2"] "2-2852-2852.1195-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.1195" [] [[0.0, 0.0]] 0 true true false false 0.46904212338343926 0 1.88947159874052019691734751932 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1195/1"] "2-2852-2852.1379-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.1379" [] [[0.0, 0.0]] 0 true true false false -0.35364727042335864 0 0.70792727096024485940017157323 ["ModularForm/GL2/Q/holomorphic/2852/1/r/b/1379/1"] "2-2852-2852.1379-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.1379" [] [[0.0, 0.0]] 0 true true false false 0.04635272957664134 0 0.816904262518368607769223700373 ["ModularForm/GL2/Q/holomorphic/2852/1/r/a/1379/1"] "2-2852-2852.1379-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.1379" [] [[0.0, 0.0]] 0 true true false false -0.020313937090025328 0 1.11116198528363091496487511719 ["ModularForm/GL2/Q/holomorphic/2852/1/r/c/1379/1"] "2-2852-2852.1379-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.1379" [] [[0.0, 0.0]] 0 true true false false 0.04635272957664134 0 1.15951823649577788565442878178 ["ModularForm/GL2/Q/holomorphic/2852/1/r/d/1379/2"] "2-2852-2852.1379-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.1379" [] [[0.0, 0.0]] 0 true true false false 0.313019396243308 0 1.46866346241997097672238439857 ["ModularForm/GL2/Q/holomorphic/2852/1/r/c/1379/2"] "2-2852-2852.1379-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.1379" [] [[0.0, 0.0]] 0 true true false false 0.04635272957664134 0 1.50225151882934940670554483657 ["ModularForm/GL2/Q/holomorphic/2852/1/r/d/1379/1"] "2-2852-2852.1563-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.1563" [] [[0.0, 0.0]] 0 true true false false -0.005184169962411247 0 0.63448155906658111912593601424 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1563/2"] "2-2852-2852.1563-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.1563" [] [[0.0, 0.0]] 0 true true false false -0.29407305885130014 0 0.863818661547397339764007419993 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1563/3"] "2-2852-2852.1563-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.1563" [] [[0.0, 0.0]] 0 true true false false 0.37259360781536655 0 1.04272011280388810229005065025 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1563/2"] "2-2852-2852.1563-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.1563" [] [[0.0, 0.0]] 0 true true false false -0.005184169962411247 0 1.10227383437927982856851858685 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1563/1"] "2-2852-2852.1563-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.1563" [] [[0.0, 0.0]] 0 true true false false 0.0392602744820332 0 1.48482365796527672938015329345 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1563/1"] "2-2852-2852.1563-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.1563" [] [[0.0, 0.0]] 0 true true false false -0.005184169962411247 0 1.91178196021085364230993617382 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1563/3"] "2-2852-2852.1655-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.1655" [] [[0.0, 0.0]] 0 true true false false -0.37259360781536655 0 0.29780574635424526754326663614 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1655/2"] "2-2852-2852.1655-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.1655" [] [[0.0, 0.0]] 0 true true false false 0.005184169962411247 0 0.54002706404061622543359768908 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1655/1"] "2-2852-2852.1655-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.1655" [] [[0.0, 0.0]] 0 true true false false 0.005184169962411247 0 1.26056072665839410202097190795 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1655/2"] "2-2852-2852.1655-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.1655" [] [[0.0, 0.0]] 0 true true false false -0.0392602744820332 0 1.38169979459184599269320868463 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1655/1"] "2-2852-2852.1655-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.1655" [] [[0.0, 0.0]] 0 true true false false 0.005184169962411247 0 1.82990800210942261925409063206 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1655/3"] "2-2852-2852.1655-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.1655" [] [[0.0, 0.0]] 0 true true false false 0.29407305885130014 0 1.85618650917109130610514912088 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1655/3"] "2-2852-2852.1747-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.1747" [] [[0.0, 0.0]] 0 true true false false -0.13570879005010586 0 0.19801570001201527498578969005 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1747/2"] "2-2852-2852.1747-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.1747" [] [[0.0, 0.0]] 0 true true false false -0.046819901161216974 0 0.845617904848286836025626408072 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1747/3"] "2-2852-2852.1747-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.1747" [] [[0.0, 0.0]] 0 true true false false -0.046819901161216974 0 0.887188863456246225418045694771 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1747/2"] "2-2852-2852.1747-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.1747" [] [[0.0, 0.0]] 0 true true false false -0.46904212338343926 0 0.891965326296887250153726988829 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1747/1"] "2-2852-2852.1747-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.1747" [] [[0.0, 0.0]] 0 true true false false -0.046819901161216974 0 1.31131824041825283337919271370 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/1747/1"] "2-2852-2852.1747-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.1747" [] [[0.0, 0.0]] 0 true true false false 0.19762454328322748 0 1.74474692470544804011461484853 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/1747/3"] "2-2852-2852.2207-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.2207" [] [[0.0, 0.0]] 0 true true false false 0.3499042140085687 0 0.874668885051404867486032469978 ["ModularForm/GL2/Q/holomorphic/2852/1/l/b/2207/2"] "2-2852-2852.2207-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.2207" [] [[0.0, 0.0]] 0 true true false false 0.4610153251196798 0 0.882399875025328174552458012093 ["ModularForm/GL2/Q/holomorphic/2852/1/l/a/2207/2"] "2-2852-2852.2207-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.2207" [] [[0.0, 0.0]] 0 true true false false 0.01657088067523534 0 1.26446453598403721038834358339 ["ModularForm/GL2/Q/holomorphic/2852/1/l/b/2207/1"] "2-2852-2852.2207-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.2207" [] [[0.0, 0.0]] 0 true true false false -0.31676245265809805 0 1.31544356768211153542818309897 ["ModularForm/GL2/Q/holomorphic/2852/1/l/b/2207/3"] "2-2852-2852.2207-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.2207" [] [[0.0, 0.0]] 0 true true false false 0.4610153251196798 0 1.61080016286308125074513346885 ["ModularForm/GL2/Q/holomorphic/2852/1/l/a/2207/1"] "2-2852-2852.2207-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.2207" [] [[0.0, 0.0]] 0 true true false false 0.4610153251196798 0 2.01205945765644277995884162291 ["ModularForm/GL2/Q/holomorphic/2852/1/l/a/2207/3"] "2-2852-2852.2483-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.2483" [] [[0.0, 0.0]] 0 true true false false -0.31064393952653546 0 0.62771569174629043051821667895 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/2483/1"] "2-2852-2852.2483-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.2483" [] [[0.0, 0.0]] 0 true true false false 0.022689393806797863 0 1.03790455343859261272789485816 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/2483/3"] "2-2852-2852.2483-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.2483" [] [[0.0, 0.0]] 0 true true false false 0.20046717158457564 0 1.43366749737807129697750307522 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/2483/1"] "2-2852-2852.2483-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.2483" [] [[0.0, 0.0]] 0 true true false false 0.20046717158457564 0 1.55350428032554472229139952995 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/2483/2"] "2-2852-2852.2483-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.2483" [] [[0.0, 0.0]] 0 true true false false 0.20046717158457564 0 1.63150877864876275348783100542 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/b/2483/3"] "2-2852-2852.2483-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.2483" [] [[0.0, 0.0]] 0 true true false false 0.3560227271401312 0 1.99208163356054128819433644387 ["ModularForm/GL2/Q/holomorphic/2852/1/bn/a/2483/2"] "2-2852-2852.275-c0-0-0" 1.1930353792026969 1.423333416029323 2 2852 "2852.275" [] [[0.0, 0.0]] 0 true true false false -0.3583642688011943 0 0.68680925137261035997692025309 ["ModularForm/GL2/Q/holomorphic/2852/1/r/a/275/1"] "2-2852-2852.275-c0-0-1" 1.1930353792026969 1.423333416029323 2 2852 "2852.275" [] [[0.0, 0.0]] 0 true true false false -0.3583642688011943 0 0.914112506371285508166954423050 ["ModularForm/GL2/Q/holomorphic/2852/1/r/d/275/2"] "2-2852-2852.275-c0-0-2" 1.1930353792026969 1.423333416029323 2 2852 "2852.275" [] [[0.0, 0.0]] 0 true true false false -0.22503093546786093 0 1.01068314049592677200859375699 ["ModularForm/GL2/Q/holomorphic/2852/1/r/c/275/2"] "2-2852-2852.275-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.275" [] [[0.0, 0.0]] 0 true true false false 0.10830239786547241 0 1.12171670676367907298370015111 ["ModularForm/GL2/Q/holomorphic/2852/1/r/c/275/1"] "2-2852-2852.275-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.275" [] [[0.0, 0.0]] 0 true true false false 0.4416357311988058 0 2.22266692067722078629004942571 ["ModularForm/GL2/Q/holomorphic/2852/1/r/b/275/1"] "2-2852-2852.275-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.275" [] [[0.0, 0.0]] 0 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"ModularForm/GL2/Q/holomorphic/2852/1/g/a", "ArtinRepresentation/2.2852.4t3.d.a", "ArtinRepresentation/2.2852.4t3.d"] "2-2852-2852.2851-c0-0-3" 1.1930353792026969 1.423333416029323 2 2852 "2852.2851" [] [[0.0, 0.0]] 0 true true false false -0.33333333333333337 0 0.946730781380826238048131080744 ["ModularForm/GL2/Q/holomorphic/2852/1/g/j/2851/2"] "2-2852-2852.2851-c0-0-4" 1.1930353792026969 1.423333416029323 2 2852 "2852.2851" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.981847636454672290948512053840 ["ModularForm/GL2/Q/holomorphic/2852/1/g/h/2851/1"] "2-2852-2852.2851-c0-0-5" 1.1930353792026969 1.423333416029323 2 2852 "2852.2851" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.987895731620733889804256013025 ["ModularForm/GL2/Q/holomorphic/2852/1/g/g/2851/2"] "2-2852-2852.2851-c0-0-6" 1.1930353792026969 1.423333416029323 2 2852 "2852.2851" [] [[0.0, 0.0]] 0 true true true true 0.0 0 1.07851373258340142259646976835 ["ModularForm/GL2/Q/holomorphic/2852/1/g/c/2851/1", 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1.58742808756384876592525246891 ["ModularForm/GL2/Q/holomorphic/2852/1/r/d/91/2"] # 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).