# lfunc_search downloaded from the LMFDB on 24 June 2026. # Search link: https://www.lmfdb.org/L/2/110/11.10/c2-0 # Query "{'degree': 2, 'conductor': 110, 'spectral_label': 'c2-0'}" returned 216 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. "2-110-11.10-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false -0.31557249661879 0 0.17581810111552357558762431813 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/2"] "2-110-11.10-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false -0.11431805387932414 0 0.64569573679125452796676559794 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/1"] "2-110-11.10-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false -0.38568194612067586 0 0.74216746192361817759719430994 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/7"] "2-110-11.10-c2-0-3" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false -0.18442750338121003 0 1.69901907120356083335221630695 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/8"] "2-110-11.10-c2-0-4" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false 0.11431805387932414 0 1.70528011818008952079499287072 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/5"] "2-110-11.10-c2-0-5" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false 0.31557249661879 0 2.42438123531777725463289021039 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/6"] "2-110-11.10-c2-0-6" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false 0.18442750338121003 0 2.53463521364078120574943074451 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/4"] "2-110-11.10-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "11.10" [] [[1.0, 0.0]] 2 true true false false 0.38568194612067586 0 2.61190160907450069282470983728 ["ModularForm/GL2/Q/holomorphic/110/3/d/a/21/3"] "2-110-11.2-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false -0.3867235081642835 0 0.27849069734997435232525388085 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/101/1"] "2-110-11.2-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false -0.30323878105156676 0 0.990030151549515383288200766349 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/101/3"] "2-110-11.2-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false -0.11650461686047762 0 1.05673297159839227531999473189 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/101/2"] "2-110-11.2-c2-0-3" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false 0.34893511613439426 0 1.72191079804069974840197738184 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/101/1"] "2-110-11.2-c2-0-4" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false -0.007058504157220003 0 2.25306274577149045738527945129 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/101/4"] "2-110-11.2-c2-0-5" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false 0.2166451219592685 0 2.30608076733330539696516735246 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/101/2"] "2-110-11.2-c2-0-6" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false 0.027622834028192777 0 2.35779779814329867797202906475 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/101/3"] "2-110-11.2-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "11.2" [] [[1.0, 0.0]] 2 true true false false 0.06458876624195214 0 2.92010160426835003862580117246 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/101/4"] "2-110-11.6-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false -0.34893511613439426 0 0.05851460972206457985396268785 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/61/1"] "2-110-11.6-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false -0.2166451219592685 0 1.27034303478774363503377713247 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/61/2"] "2-110-11.6-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false 0.11650461686047762 0 1.84295056013737399772684590960 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/61/2"] "2-110-11.6-c2-0-3" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false -0.027622834028192777 0 1.90479554057342348079190298846 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/61/3"] "2-110-11.6-c2-0-4" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false 0.007058504157220003 0 2.26508436933791598313713932392 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/61/4"] "2-110-11.6-c2-0-5" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false -0.06458876624195214 0 2.29647232057732964397598201692 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/61/4"] "2-110-11.6-c2-0-6" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false 0.3867235081642835 0 2.37803269102313033734347487925 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/61/1"] "2-110-11.6-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "11.6" [] [[1.0, 0.0]] 2 true true false false 0.30323878105156676 0 2.87681483824314027058362109004 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/61/3"] "2-110-11.7-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false -0.46190661593379984 0 0.35336565911475357708510860554 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/51/1"] "2-110-11.7-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false -0.3442983900519778 0 0.61167426650132210291944510592 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/51/3"] "2-110-11.7-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false -0.12594949341562336 0 1.44039662789522830895469081336 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/51/1"] "2-110-11.7-c2-0-3" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false -0.020228624653069953 0 1.63901200679936843454819177777 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/51/2"] "2-110-11.7-c2-0-4" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false 0.06387137350737324 0 1.87071004303584131978056194172 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/51/3"] "2-110-11.7-c2-0-5" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false 0.2413053673498219 0 2.70864614453000811538888779197 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/51/2"] "2-110-11.7-c2-0-6" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false 0.31943536498240466 0 3.09334053547550002315170861106 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/51/4"] "2-110-11.7-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "11.7" [] [[1.0, 0.0]] 2 true true false false 0.3137744869019853 0 3.17795239831438293053415798248 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/51/4"] "2-110-11.8-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false -0.2413053673498219 0 1.02710370212829691695628491559 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/41/2"] "2-110-11.8-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false -0.31943536498240466 0 1.38253377962203862260704563570 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/41/4"] "2-110-11.8-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false 0.020228624653069953 0 1.43281717640845314740919067396 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/41/2"] "2-110-11.8-c2-0-3" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false -0.3137744869019853 0 1.44325311545267255611175668618 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/41/4"] "2-110-11.8-c2-0-4" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false 0.12594949341562336 0 1.68654443943407795532145367362 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/41/1"] "2-110-11.8-c2-0-5" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false -0.06387137350737324 0 1.74183435815319961562257908616 ["ModularForm/GL2/Q/holomorphic/110/3/h/a/41/3"] "2-110-11.8-c2-0-6" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false 0.46190661593379984 0 2.72357811777457043330309199466 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/41/1"] "2-110-11.8-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "11.8" [] [[1.0, 0.0]] 2 true true false false 0.3442983900519778 0 3.25297796332941572308421516510 ["ModularForm/GL2/Q/holomorphic/110/3/h/b/41/3"] "2-110-5.2-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false 0.42471279031339637 0 0.38139672515141841582486898721 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/67/1"] "2-110-5.2-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false -0.413317986251086 0 0.77577281628209763267261630352 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/67/2"] "2-110-5.2-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false -0.23598476062745322 0 1.25413725488181813446046053650 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/67/4"] "2-110-5.2-c2-0-3" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false -0.02659896163332544 0 1.27925040019813699091949898679 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/67/2"] "2-110-5.2-c2-0-4" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false -0.21860685994408735 0 1.43894894715655113747204811874 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/67/3"] "2-110-5.2-c2-0-5" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false 0.415817286573609 0 2.00207878440988130041452915395 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/67/1"] "2-110-5.2-c2-0-6" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false -0.03623895584938569 0 2.18783490760455290195194653068 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/67/6"] "2-110-5.2-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false 0.21715378353370599 0 2.37313480445023955075761057583 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/67/4"] "2-110-5.2-c2-0-8" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false 0.05322758118383252 0 2.68976720243431459925110977902 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/67/5"] "2-110-5.2-c2-0-9" 1.7312662729815036 2.997282907963266 2 110 "5.2" [] [[1.0, 0.0]] 2 true true false false 0.39362789152601046 0 2.93727460798711395188889751351 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/67/3"] "2-110-5.3-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false -0.415817286573609 0 0.05029525597203157810742850958 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/23/1"] "2-110-5.3-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false -0.39362789152601046 0 0.852553784533122064547512714605 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/23/3"] "2-110-5.3-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false -0.21715378353370599 0 1.82373167399978654669860249350 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/23/4"] "2-110-5.3-c2-0-3" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false -0.05322758118383252 0 1.82636353867444018528619339270 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/23/5"] "2-110-5.3-c2-0-4" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false 0.02659896163332544 0 1.93523182052028076872275240459 ["ModularForm/GL2/Q/holomorphic/110/3/e/a/23/2"] "2-110-5.3-c2-0-5" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false 0.03623895584938569 0 2.30531038975903980125358603785 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/23/6"] "2-110-5.3-c2-0-6" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false 0.21860685994408735 0 2.47267247873565288826495939873 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/23/3"] "2-110-5.3-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false 0.23598476062745322 0 2.58462746346738766818490572242 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/23/4"] "2-110-5.3-c2-0-8" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false 0.413317986251086 0 3.31556155518412269417980149092 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/23/2"] "2-110-5.3-c2-0-9" 1.7312662729815036 2.997282907963266 2 110 "5.3" [] [[1.0, 0.0]] 2 true true false false -0.42471279031339637 0 3.98235929796484570141550405456 ["ModularForm/GL2/Q/holomorphic/110/3/e/b/23/1"] "2-110-55.19-c2-0-0" 1.7312662729815036 2.997282907963266 2 110 "55.19" [] [[1.0, 0.0]] 2 true true false false -0.4050608275798364 0 0.13666068843083355990993600004 ["ModularForm/GL2/Q/holomorphic/110/3/i/a/19/8"] "2-110-55.19-c2-0-1" 1.7312662729815036 2.997282907963266 2 110 "55.19" [] [[1.0, 0.0]] 2 true true false false -0.40476816830585244 0 0.43969835517404238040657477301 ["ModularForm/GL2/Q/holomorphic/110/3/i/a/19/4"] "2-110-55.19-c2-0-10" 1.7312662729815036 2.997282907963266 2 110 "55.19" [] [[1.0, 0.0]] 2 true true false false 0.1526941034077727 0 2.80036416509586210934581042779 ["ModularForm/GL2/Q/holomorphic/110/3/i/a/19/11"] "2-110-55.19-c2-0-11" 1.7312662729815036 2.997282907963266 2 110 "55.19" [] [[1.0, 0.0]] 2 true true false false 0.1429318559617745 0 2.84097679211148819137707620660 ["ModularForm/GL2/Q/holomorphic/110/3/i/a/19/12"] "2-110-55.19-c2-0-2" 1.7312662729815036 2.997282907963266 2 110 "55.19" [] [[1.0, 0.0]] 2 true true false false -0.40482211420295433 0 0.55039611778636830107756627974 ["ModularForm/GL2/Q/holomorphic/110/3/i/a/19/3"] "2-110-55.19-c2-0-3" 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2.997282907963266 2 110 "55.54" [] [[1.0, 0.0]] 2 true true false false -0.004228701613190135 0 1.96224146494550813878334099239 ["ModularForm/GL2/Q/holomorphic/110/3/c/b/109/6"] "2-110-55.54-c2-0-7" 1.7312662729815036 2.997282907963266 2 110 "55.54" [] [[1.0, 0.0]] 2 true true false false 0.004228701613190135 0 2.13142185177530060207208006443 ["ModularForm/GL2/Q/holomorphic/110/3/c/b/109/7"] "2-110-55.54-c2-0-8" 1.7312662729815036 2.997282907963266 2 110 "55.54" [] [[1.0, 0.0]] 2 true true false false 0.2790991158233053 0 2.34855318429310490678892123359 ["ModularForm/GL2/Q/holomorphic/110/3/c/a/109/1"] "2-110-55.54-c2-0-9" 1.7312662729815036 2.997282907963266 2 110 "55.54" [] [[1.0, 0.0]] 2 true true false false 0.23146877921898779 0 2.98923183501691408625318562296 ["ModularForm/GL2/Q/holomorphic/110/3/c/b/109/5"] # 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).