# lfunc_search downloaded from the LMFDB on 21 September 2026. # Search link: https://www.lmfdb.org/L/2/1682 # Query "{'degree': 2, 'conductor': 1682}" returned 339 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. "2-1682-1.1-c1-0-0" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.29041457591471203478190389304 ["ModularForm/GL2/Q/holomorphic/1682/2/a/u/1/5"] "2-1682-1.1-c1-0-1" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.38124474427974271979000169450 ["EllipticCurve/Q/1682/b", "ModularForm/GL2/Q/holomorphic/1682/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/b"] "2-1682-1.1-c1-0-10" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.78027801002833452227862802415 ["ModularForm/GL2/Q/holomorphic/1682/2/a/v/1/2"] "2-1682-1.1-c1-0-11" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.816413476773546080974234511699 ["ModularForm/GL2/Q/holomorphic/1682/2/a/r/1/3"] "2-1682-1.1-c1-0-12" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.820504771086699217743416607845 ["ModularForm/GL2/Q/holomorphic/1682/2/a/k/1/1"] "2-1682-1.1-c1-0-13" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.960087605343476753715430007341 ["ModularForm/GL2/Q/holomorphic/1682/2/a/u/1/3"] "2-1682-1.1-c1-0-14" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.972419309871331486021568763471 ["ModularForm/GL2/Q/holomorphic/1682/2/a/t/1/2"] "2-1682-1.1-c1-0-15" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.00206100855434364054687831081 ["ModularForm/GL2/Q/holomorphic/1682/2/a/r/1/5"] "2-1682-1.1-c1-0-16" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.02185728567262492076209475344 ["EllipticCurve/Q/1682/a", "ModularForm/GL2/Q/holomorphic/1682/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/a"] "2-1682-1.1-c1-0-17" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.03759100897086352742471870949 ["EllipticCurve/Q/1682/f", "ModularForm/GL2/Q/holomorphic/1682/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/f"] "2-1682-1.1-c1-0-18" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.04138971571346308880173133745 ["ModularForm/GL2/Q/holomorphic/1682/2/a/o/1/1"] "2-1682-1.1-c1-0-19" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.07000927913891941226693964243 ["ModularForm/GL2/Q/holomorphic/1682/2/a/u/1/2"] "2-1682-1.1-c1-0-2" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.43144112809964217539602525856 ["ModularForm/GL2/Q/holomorphic/1682/2/a/t/1/1"] "2-1682-1.1-c1-0-20" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11451182319642144438661893140 ["ModularForm/GL2/Q/holomorphic/1682/2/a/r/1/4"] "2-1682-1.1-c1-0-21" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11712543032732563297561577995 ["ModularForm/GL2/Q/holomorphic/1682/2/a/v/1/5"] "2-1682-1.1-c1-0-22" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.12305800056014304267753887827 ["ModularForm/GL2/Q/holomorphic/1682/2/a/k/1/2"] "2-1682-1.1-c1-0-23" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.13890251403859473527163969507 ["ModularForm/GL2/Q/holomorphic/1682/2/a/v/1/1"] "2-1682-1.1-c1-0-24" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.17419081584039026599769630241 ["ModularForm/GL2/Q/holomorphic/1682/2/a/n/1/1"] "2-1682-1.1-c1-0-25" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.19884447357088848587887105758 ["ModularForm/GL2/Q/holomorphic/1682/2/a/q/1/1"] "2-1682-1.1-c1-0-26" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.20531600057351602568872723777 ["ModularForm/GL2/Q/holomorphic/1682/2/a/n/1/2"] "2-1682-1.1-c1-0-27" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.20685603464340185344773931194 ["ModularForm/GL2/Q/holomorphic/1682/2/a/u/1/7"] "2-1682-1.1-c1-0-28" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.30247724091888463411313228975 ["ModularForm/GL2/Q/holomorphic/1682/2/a/v/1/3"] "2-1682-1.1-c1-0-29" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.32364216049448157109471695089 ["EllipticCurve/Q/1682/j", "ModularForm/GL2/Q/holomorphic/1682/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/j"] "2-1682-1.1-c1-0-3" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.45563952397857100426204094077 ["ModularForm/GL2/Q/holomorphic/1682/2/a/r/1/1"] "2-1682-1.1-c1-0-30" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.39029679678055849977210783834 ["ModularForm/GL2/Q/holomorphic/1682/2/a/s/1/1"] "2-1682-1.1-c1-0-31" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.39084613196056918568525693843 ["ModularForm/GL2/Q/holomorphic/1682/2/a/p/1/1"] "2-1682-1.1-c1-0-32" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.39294319804184584944409279177 ["ModularForm/GL2/Q/holomorphic/1682/2/a/q/1/2"] "2-1682-1.1-c1-0-33" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.39345694576822601394248550654 ["EllipticCurve/Q/1682/i", "ModularForm/GL2/Q/holomorphic/1682/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/i"] "2-1682-1.1-c1-0-34" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.40185954049603372478493528134 ["ModularForm/GL2/Q/holomorphic/1682/2/a/r/1/6"] "2-1682-1.1-c1-0-35" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.44038511736397468392249627681 ["ModularForm/GL2/Q/holomorphic/1682/2/a/l/1/1"] "2-1682-1.1-c1-0-36" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.44226286578634658557135139241 ["ModularForm/GL2/Q/holomorphic/1682/2/a/u/1/6"] "2-1682-1.1-c1-0-37" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.44255540990037051821560921801 ["ModularForm/GL2/Q/holomorphic/1682/2/a/u/1/8"] "2-1682-1.1-c1-0-38" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.47593652820112294085842230992 ["ModularForm/GL2/Q/holomorphic/1682/2/a/q/1/3"] "2-1682-1.1-c1-0-39" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.48630710283875076660165321951 ["ModularForm/GL2/Q/holomorphic/1682/2/a/m/1/2"] "2-1682-1.1-c1-0-4" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.56123636351846993743922430141 ["ModularForm/GL2/Q/holomorphic/1682/2/a/u/1/1"] "2-1682-1.1-c1-0-40" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.50115772037141317240872216813 ["EllipticCurve/Q/1682/e", "ModularForm/GL2/Q/holomorphic/1682/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/e"] "2-1682-1.1-c1-0-41" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.50784832396034611554159383313 ["ModularForm/GL2/Q/holomorphic/1682/2/a/o/1/2"] "2-1682-1.1-c1-0-42" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.52739268380421819658494041711 ["ModularForm/GL2/Q/holomorphic/1682/2/a/v/1/8"] "2-1682-1.1-c1-0-43" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.55007135648515435454103344133 ["ModularForm/GL2/Q/holomorphic/1682/2/a/m/1/1"] "2-1682-1.1-c1-0-44" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.55213401277962898809576348988 ["ModularForm/GL2/Q/holomorphic/1682/2/a/q/1/4"] "2-1682-1.1-c1-0-45" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.55501546504386044257450298236 ["ModularForm/GL2/Q/holomorphic/1682/2/a/t/1/5"] "2-1682-1.1-c1-0-46" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.56025530297488037727488772479 ["ModularForm/GL2/Q/holomorphic/1682/2/a/t/1/4"] "2-1682-1.1-c1-0-47" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.59626437970559016349034665980 ["ModularForm/GL2/Q/holomorphic/1682/2/a/l/1/2"] "2-1682-1.1-c1-0-48" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.64898882454024260621104157059 ["ModularForm/GL2/Q/holomorphic/1682/2/a/s/1/3"] "2-1682-1.1-c1-0-49" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.66981753705387663269958363527 ["ModularForm/GL2/Q/holomorphic/1682/2/a/n/1/3"] "2-1682-1.1-c1-0-5" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.61439046057345923085639409537 ["EllipticCurve/Q/1682/c", "ModularForm/GL2/Q/holomorphic/1682/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/c"] "2-1682-1.1-c1-0-50" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.71057913370762712471889035449 ["ModularForm/GL2/Q/holomorphic/1682/2/a/v/1/7"] "2-1682-1.1-c1-0-51" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.71861950252158898504970163132 ["ModularForm/GL2/Q/holomorphic/1682/2/a/v/1/6"] "2-1682-1.1-c1-0-52" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.73149035395535244117517059817 ["ModularForm/GL2/Q/holomorphic/1682/2/a/q/1/6"] "2-1682-1.1-c1-0-53" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.74984707092054903729351227527 ["ModularForm/GL2/Q/holomorphic/1682/2/a/s/1/2"] "2-1682-1.1-c1-0-54" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.76909934994324200300399553047 ["EllipticCurve/Q/1682/d", "ModularForm/GL2/Q/holomorphic/1682/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/d"] "2-1682-1.1-c1-0-55" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.82178337192159371048211739735 ["ModularForm/GL2/Q/holomorphic/1682/2/a/s/1/4"] "2-1682-1.1-c1-0-56" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.83462916634357843541704975263 ["ModularForm/GL2/Q/holomorphic/1682/2/a/t/1/6"] "2-1682-1.1-c1-0-57" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.85595880764523973402155825677 ["ModularForm/GL2/Q/holomorphic/1682/2/a/m/1/3"] "2-1682-1.1-c1-0-58" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.89199683662013500547491029070 ["ModularForm/GL2/Q/holomorphic/1682/2/a/o/1/3"] "2-1682-1.1-c1-0-59" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.90114708353051726091892910819 ["ModularForm/GL2/Q/holomorphic/1682/2/a/s/1/5"] "2-1682-1.1-c1-0-6" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.67570735199855790472226398270 ["ModularForm/GL2/Q/holomorphic/1682/2/a/r/1/2"] "2-1682-1.1-c1-0-60" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.98380809317553696143260797903 ["ModularForm/GL2/Q/holomorphic/1682/2/a/o/1/4"] "2-1682-1.1-c1-0-61" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 2.02901758864353404212408232666 ["ModularForm/GL2/Q/holomorphic/1682/2/a/p/1/2"] "2-1682-1.1-c1-0-62" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 2.12178238161665781855669022134 ["ModularForm/GL2/Q/holomorphic/1682/2/a/p/1/3"] "2-1682-1.1-c1-0-63" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 2.12584750664501524964339482267 ["ModularForm/GL2/Q/holomorphic/1682/2/a/s/1/6"] "2-1682-1.1-c1-0-64" 3.664810720893968 13.430837619979368 2 1682 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 2.15710567472534577568345191269 ["EllipticCurve/Q/1682/g", "ModularForm/GL2/Q/holomorphic/1682/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/1682/2/a/g"] "2-1682-1.1-c1-0-65" 3.664810720893968 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0.15790616577621384799838680340 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/10"] "2-1682-1.1-c3-0-40" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.55344269026558083189075316177 ["ModularForm/GL2/Q/holomorphic/1682/4/a/w/1/8"] "2-1682-1.1-c3-0-41" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.55454160618978957115586958776 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/5"] "2-1682-1.1-c3-0-42" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.56899872998062809007579429815 ["ModularForm/GL2/Q/holomorphic/1682/4/a/n/1/5"] "2-1682-1.1-c3-0-43" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.58448118303594626672102544256 ["ModularForm/GL2/Q/holomorphic/1682/4/a/o/1/1"] "2-1682-1.1-c3-0-44" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.59969488998899732112033303529 ["ModularForm/GL2/Q/holomorphic/1682/4/a/u/1/12"] "2-1682-1.1-c3-0-45" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.60103402777369654792600096179 ["ModularForm/GL2/Q/holomorphic/1682/4/a/g/1/3"] "2-1682-1.1-c3-0-46" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.62618183724291235230671818773 ["ModularForm/GL2/Q/holomorphic/1682/4/a/q/1/1"] "2-1682-1.1-c3-0-47" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.62712041912088856929294464600 ["ModularForm/GL2/Q/holomorphic/1682/4/a/u/1/3"] "2-1682-1.1-c3-0-48" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.63781402278971101595875072042 ["ModularForm/GL2/Q/holomorphic/1682/4/a/v/1/12"] "2-1682-1.1-c3-0-49" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.66557919876193425958115983742 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/4"] "2-1682-1.1-c3-0-5" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.18012986549109359424214815288 ["ModularForm/GL2/Q/holomorphic/1682/4/a/o/1/5"] "2-1682-1.1-c3-0-50" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.66606083457973728710655068278 ["ModularForm/GL2/Q/holomorphic/1682/4/a/q/1/6"] "2-1682-1.1-c3-0-51" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.66951474518161935117106181676 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/2"] "2-1682-1.1-c3-0-52" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.69523914402957793823413481584 ["ModularForm/GL2/Q/holomorphic/1682/4/a/o/1/7"] "2-1682-1.1-c3-0-53" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.70197431547996578228419687740 ["ModularForm/GL2/Q/holomorphic/1682/4/a/r/1/5"] "2-1682-1.1-c3-0-54" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.70834167204090784032643655430 ["ModularForm/GL2/Q/holomorphic/1682/4/a/u/1/10"] "2-1682-1.1-c3-0-55" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.71298487692098207342523869988 ["ModularForm/GL2/Q/holomorphic/1682/4/a/m/1/6"] "2-1682-1.1-c3-0-56" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.71749763189985703077172652790 ["ModularForm/GL2/Q/holomorphic/1682/4/a/k/1/2"] "2-1682-1.1-c3-0-57" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.74013384393161317376467212314 ["ModularForm/GL2/Q/holomorphic/1682/4/a/m/1/7"] "2-1682-1.1-c3-0-58" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.77371917599544924784764239408 ["ModularForm/GL2/Q/holomorphic/1682/4/a/v/1/7"] "2-1682-1.1-c3-0-59" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.77431870548800577781327727150 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/15"] "2-1682-1.1-c3-0-6" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.19269835435208108866890710232 ["ModularForm/GL2/Q/holomorphic/1682/4/a/u/1/6"] "2-1682-1.1-c3-0-60" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.77876767954733537149846389474 ["ModularForm/GL2/Q/holomorphic/1682/4/a/i/1/2"] "2-1682-1.1-c3-0-61" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.77978684216387226307207981480 ["ModularForm/GL2/Q/holomorphic/1682/4/a/s/1/2"] "2-1682-1.1-c3-0-62" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.78141652051755239395628229595 ["ModularForm/GL2/Q/holomorphic/1682/4/a/n/1/2"] "2-1682-1.1-c3-0-63" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.798433930422556352078010034678 ["ModularForm/GL2/Q/holomorphic/1682/4/a/k/1/1"] "2-1682-1.1-c3-0-64" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.799418816490740785271016058343 ["ModularForm/GL2/Q/holomorphic/1682/4/a/n/1/1"] "2-1682-1.1-c3-0-65" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.813304836283482798319797246658 ["ModularForm/GL2/Q/holomorphic/1682/4/a/o/1/3"] "2-1682-1.1-c3-0-66" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.814155752454691207192708028648 ["ModularForm/GL2/Q/holomorphic/1682/4/a/u/1/16"] "2-1682-1.1-c3-0-67" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.815852275031129351739959447864 ["ModularForm/GL2/Q/holomorphic/1682/4/a/v/1/4"] "2-1682-1.1-c3-0-68" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.819906837412986712104187136620 ["ModularForm/GL2/Q/holomorphic/1682/4/a/q/1/4"] "2-1682-1.1-c3-0-69" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.822573296573135669478882789345 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/1"] "2-1682-1.1-c3-0-7" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.20319599185861924423659887905 ["ModularForm/GL2/Q/holomorphic/1682/4/a/q/1/2"] "2-1682-1.1-c3-0-70" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.823379720826262272851401762042 ["ModularForm/GL2/Q/holomorphic/1682/4/a/v/1/11"] "2-1682-1.1-c3-0-71" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.825410330117697681638413728857 ["ModularForm/GL2/Q/holomorphic/1682/4/a/v/1/10"] "2-1682-1.1-c3-0-72" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.828625326811289046816273762430 ["ModularForm/GL2/Q/holomorphic/1682/4/a/q/1/10"] "2-1682-1.1-c3-0-73" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.829703096923132273318901754510 ["ModularForm/GL2/Q/holomorphic/1682/4/a/g/1/1"] "2-1682-1.1-c3-0-74" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.835258117170893640231036586836 ["ModularForm/GL2/Q/holomorphic/1682/4/a/j/1/2"] "2-1682-1.1-c3-0-75" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.844590485620938356805025819638 ["ModularForm/GL2/Q/holomorphic/1682/4/a/o/1/6"] "2-1682-1.1-c3-0-76" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.845405375725774836414453785037 ["ModularForm/GL2/Q/holomorphic/1682/4/a/u/1/1"] "2-1682-1.1-c3-0-77" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.846616281675625739886459651277 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/11"] "2-1682-1.1-c3-0-78" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.852041947192570515363422284819 ["ModularForm/GL2/Q/holomorphic/1682/4/a/q/1/5"] "2-1682-1.1-c3-0-79" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.854259883552544698143164547792 ["ModularForm/GL2/Q/holomorphic/1682/4/a/w/1/15"] "2-1682-1.1-c3-0-8" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.25586774872388556380574367014 ["ModularForm/GL2/Q/holomorphic/1682/4/a/g/1/2"] "2-1682-1.1-c3-0-80" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.854852287212110706326638098599 ["ModularForm/GL2/Q/holomorphic/1682/4/a/w/1/10"] "2-1682-1.1-c3-0-81" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.857483546598266677917909721201 ["ModularForm/GL2/Q/holomorphic/1682/4/a/r/1/3"] "2-1682-1.1-c3-0-82" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.868690423811677816895396700791 ["ModularForm/GL2/Q/holomorphic/1682/4/a/m/1/3"] "2-1682-1.1-c3-0-83" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.876562991687717096545461167452 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/21"] "2-1682-1.1-c3-0-84" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.891614494789029466126363787243 ["ModularForm/GL2/Q/holomorphic/1682/4/a/r/1/1"] "2-1682-1.1-c3-0-85" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.898891242651858589354959811109 ["ModularForm/GL2/Q/holomorphic/1682/4/a/x/1/3"] "2-1682-1.1-c3-0-86" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.902182190778053242972817573732 ["ModularForm/GL2/Q/holomorphic/1682/4/a/p/1/2"] "2-1682-1.1-c3-0-87" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.902860220920909441800957181441 ["ModularForm/GL2/Q/holomorphic/1682/4/a/u/1/11"] "2-1682-1.1-c3-0-88" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.905865579228775152519660069671 ["ModularForm/GL2/Q/holomorphic/1682/4/a/h/1/1"] "2-1682-1.1-c3-0-89" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.921452071961581873369552491565 ["ModularForm/GL2/Q/holomorphic/1682/4/a/r/1/2"] "2-1682-1.1-c3-0-9" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.25721272884617672985683667814 ["ModularForm/GL2/Q/holomorphic/1682/4/a/i/1/1"] "2-1682-1.1-c3-0-90" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.950053685573005657741526165956 ["ModularForm/GL2/Q/holomorphic/1682/4/a/w/1/3"] "2-1682-1.1-c3-0-91" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.952846076728319423395737863511 ["ModularForm/GL2/Q/holomorphic/1682/4/a/e/1/2"] "2-1682-1.1-c3-0-92" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.953485272328002570976735795062 ["ModularForm/GL2/Q/holomorphic/1682/4/a/w/1/12"] "2-1682-1.1-c3-0-93" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.957137118596679659287219545526 ["ModularForm/GL2/Q/holomorphic/1682/4/a/v/1/8"] "2-1682-1.1-c3-0-94" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.973025735140524734597054224411 ["ModularForm/GL2/Q/holomorphic/1682/4/a/s/1/1"] "2-1682-1.1-c3-0-95" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 0.974857809012607268298391302927 ["ModularForm/GL2/Q/holomorphic/1682/4/a/w/1/1"] "2-1682-1.1-c3-0-96" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.00372694684997197862554939543 ["ModularForm/GL2/Q/holomorphic/1682/4/a/n/1/8"] "2-1682-1.1-c3-0-97" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.00915888157971417358724315380 ["ModularForm/GL2/Q/holomorphic/1682/4/a/k/1/4"] "2-1682-1.1-c3-0-98" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.01277301255077955006065646637 ["ModularForm/GL2/Q/holomorphic/1682/4/a/e/1/1"] "2-1682-1.1-c3-0-99" 9.961988387347967 99.24121262965576 2 1682 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.01381734545836960492253857226 ["ModularForm/GL2/Q/holomorphic/1682/4/a/v/1/6"] # 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).