# lfunc_search downloaded from the LMFDB on 16 April 2026. # Search link: https://www.lmfdb.org/L/2/3328/13.6 # Query "{'degree': 2, 'conductor': 3328}" returned 304 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. "2-3328-104.21-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.21" [] [[0.0, 0.0]] 0 true true false false -0.0782082395472503 0 0.980288720884505596464552726733 ["ModularForm/GL2/Q/holomorphic/3328/1/j/a/385/1"] "2-3328-104.21-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.21" [] [[0.0, 0.0]] 0 true true false false 0.17179176045274974 0 1.65252424423980166848688918642 ["ModularForm/GL2/Q/holomorphic/3328/1/j/b/385/1", "ArtinRepresentation/2.3328.8t17.a.b"] "2-3328-104.3-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.3" [] [[0.0, 0.0]] 0 true true false false -0.16768714236295001 0 0.39421332947936960694720045424 ["ModularForm/GL2/Q/holomorphic/3328/1/v/a/1407/1"] "2-3328-104.3-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.3" [] [[0.0, 0.0]] 0 true true false false 0.08231285763704999 0 1.05625640460869781486493385536 ["ModularForm/GL2/Q/holomorphic/3328/1/v/a/1407/2"] "2-3328-104.3-c0-0-2" 1.2887545778939573 1.660888362042632 2 3328 "104.3" [] [[0.0, 0.0]] 0 true true false false -0.16768714236295001 0 1.09881127569393785392347911417 ["ModularForm/GL2/Q/holomorphic/3328/1/v/c/1407/2"] "2-3328-104.3-c0-0-3" 1.2887545778939573 1.660888362042632 2 3328 "104.3" [] [[0.0, 0.0]] 0 true true false false -0.0010204756962833475 0 1.23282429344926911721284320887 ["ModularForm/GL2/Q/holomorphic/3328/1/v/b/1407/2", "ArtinRepresentation/2.3328.24t65.a.c"] "2-3328-104.3-c0-0-4" 1.2887545778939573 1.660888362042632 2 3328 "104.3" [] [[0.0, 0.0]] 0 true true false false 0.08231285763704999 0 1.41251464315648280897802188531 ["ModularForm/GL2/Q/holomorphic/3328/1/v/c/1407/1"] "2-3328-104.3-c0-0-5" 1.2887545778939573 1.660888362042632 2 3328 "104.3" [] [[0.0, 0.0]] 0 true true false false 0.24897952430371667 0 1.56599502301287461660661888649 ["ModularForm/GL2/Q/holomorphic/3328/1/v/b/1407/1", "ArtinRepresentation/2.3328.24t65.a.a"] "2-3328-104.35-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.35" [] [[0.0, 0.0]] 0 true true false false -0.24897952430371667 0 0.72639726108826262235582461095 ["ModularForm/GL2/Q/holomorphic/3328/1/v/b/2687/2", "ArtinRepresentation/2.3328.24t65.a.d"] "2-3328-104.35-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.35" [] [[0.0, 0.0]] 0 true true false false -0.08231285763704999 0 0.73611963208662460745130436822 ["ModularForm/GL2/Q/holomorphic/3328/1/v/c/2687/1"] "2-3328-104.35-c0-0-2" 1.2887545778939573 1.660888362042632 2 3328 "104.35" [] [[0.0, 0.0]] 0 true true false false 0.0010204756962833475 0 1.03739681053479753074257913500 ["ModularForm/GL2/Q/holomorphic/3328/1/v/b/2687/1", "ArtinRepresentation/2.3328.24t65.a.b"] "2-3328-104.35-c0-0-3" 1.2887545778939573 1.660888362042632 2 3328 "104.35" [] [[0.0, 0.0]] 0 true true false false -0.08231285763704999 0 1.12827592633435009775254966264 ["ModularForm/GL2/Q/holomorphic/3328/1/v/a/2687/2"] "2-3328-104.35-c0-0-4" 1.2887545778939573 1.660888362042632 2 3328 "104.35" [] [[0.0, 0.0]] 0 true true false false 0.16768714236295001 0 1.39208863363719946902491557056 ["ModularForm/GL2/Q/holomorphic/3328/1/v/a/2687/1"] "2-3328-104.35-c0-0-5" 1.2887545778939573 1.660888362042632 2 3328 "104.35" [] [[0.0, 0.0]] 0 true true false false 0.16768714236295001 0 1.42684198494762525320702733547 ["ModularForm/GL2/Q/holomorphic/3328/1/v/c/2687/2"] "2-3328-104.37-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.37" [] [[0.0, 0.0]] 0 true true false false 0.45218776385096693 0 0.20494140139660040004872974889 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/a/2689/1"] "2-3328-104.37-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.37" [] [[0.0, 0.0]] 0 true true false false 0.20218776385096696 0 1.62234241986152846072797458785 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/b/2689/1"] "2-3328-104.43-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.43" [] [[0.0, 0.0]] 0 true true false false 0.20629238194076663 0 1.13546622096419668590799659653 ["ModularForm/GL2/Q/holomorphic/3328/1/x/a/2175/1"] "2-3328-104.43-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.43" [] [[0.0, 0.0]] 0 true true false false -0.043707618059233365 0 1.31449899475662082895194748476 ["ModularForm/GL2/Q/holomorphic/3328/1/x/a/2175/2"] "2-3328-104.45-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.45" [] [[0.0, 0.0]] 0 true true false false -0.20218776385096696 0 0.867005665124100306582583730909 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/b/2177/1"] "2-3328-104.45-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.45" [] [[0.0, 0.0]] 0 true true false false -0.45218776385096693 0 2.02349332797836632762989314853 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/a/2177/1"] "2-3328-104.5-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.5" [] [[0.0, 0.0]] 0 true true false false -0.17179176045274974 0 0.964470199143662679794386037296 ["ModularForm/GL2/Q/holomorphic/3328/1/j/b/1409/1", "ArtinRepresentation/2.3328.8t17.a.a"] "2-3328-104.5-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.5" [] [[0.0, 0.0]] 0 true true false false 0.0782082395472503 0 1.06768287165927724189489965434 ["ModularForm/GL2/Q/holomorphic/3328/1/j/a/1409/1"] "2-3328-104.51-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.51" [] [[0.0, 0.0]] 0 true true false false -0.375 0 0.26082158826684361493157156579 ["ModularForm/GL2/Q/holomorphic/3328/1/h/a/1663/2", "ArtinRepresentation/2.3328.8t11.e.a"] "2-3328-104.51-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.51" [] [[0.0, 0.0]] 0 true true false false 0.375 0 1.71886264181862685897032477003 ["ModularForm/GL2/Q/holomorphic/3328/1/h/a/1663/1", "ArtinRepresentation/2.3328.8t11.e.b"] "2-3328-104.75-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.75" [] [[0.0, 0.0]] 0 true true false false -0.20629238194076663 0 0.44609530053927863753975185683 ["ModularForm/GL2/Q/holomorphic/3328/1/x/a/127/1"] "2-3328-104.75-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.75" [] [[0.0, 0.0]] 0 true true false false 0.043707618059233365 0 1.53952168311849149372087776395 ["ModularForm/GL2/Q/holomorphic/3328/1/x/a/127/2"] "2-3328-104.85-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.85" [] [[0.0, 0.0]] 0 true true false false -0.04577128475646635 0 0.892198906343324535952034737952 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/b/2945/1"] "2-3328-104.85-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.85" [] [[0.0, 0.0]] 0 true true false false 0.20422871524353367 0 1.55275188061677362922692401739 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/a/2945/1"] "2-3328-104.93-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "104.93" [] [[0.0, 0.0]] 0 true true false false -0.20422871524353367 0 0.76009050882679437910665504439 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/a/2433/1"] "2-3328-104.93-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "104.93" [] [[0.0, 0.0]] 0 true true false false 0.04577128475646635 0 1.02724387459941760741150787824 ["ModularForm/GL2/Q/holomorphic/3328/1/bv/b/2433/1"] "2-3328-13.11-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "13.11" [] [[0.0, 0.0]] 0 true true false false 0.07718776385096696 0 1.17048844035441946714802455681 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/b/1025/1"] "2-3328-13.11-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "13.11" [] [[0.0, 0.0]] 0 true true false false 0.07718776385096696 0 1.34557565107883934846665951597 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/a/1025/1"] "2-3328-13.2-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "13.2" [] [[0.0, 0.0]] 0 true true false false 0.17077128475646636 0 1.22930708743613414738356180838 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/a/769/1"] "2-3328-13.2-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "13.2" [] [[0.0, 0.0]] 0 true true false false 0.17077128475646636 0 1.66692955712819252594880612619 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/b/769/1"] "2-3328-13.5-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "13.5" [] [[0.0, 0.0]] 0 true true false false -0.42179176045274974 0 0.18988098491569389653830170019 ["ModularForm/GL2/Q/holomorphic/3328/1/t/d/3073/1"] "2-3328-13.5-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "13.5" [] [[0.0, 0.0]] 0 true true false false -0.42179176045274974 0 0.41622736699914386261097607715 ["ModularForm/GL2/Q/holomorphic/3328/1/t/c/3073/1"] "2-3328-13.5-c0-0-2" 1.2887545778939573 1.660888362042632 2 3328 "13.5" [] [[0.0, 0.0]] 0 true true false false 0.0782082395472503 0 0.991848729997494632635025186410 ["ModularForm/GL2/Q/holomorphic/3328/1/t/c/3073/2"] "2-3328-13.5-c0-0-3" 1.2887545778939573 1.660888362042632 2 3328 "13.5" [] [[0.0, 0.0]] 0 true true false false 0.2032082395472503 0 1.18954866088576765547196896422 ["ModularForm/GL2/Q/holomorphic/3328/1/t/a/3073/1"] "2-3328-13.5-c0-0-4" 1.2887545778939573 1.660888362042632 2 3328 "13.5" [] [[0.0, 0.0]] 0 true true false false 0.0782082395472503 0 1.59938138577273879508913627901 ["ModularForm/GL2/Q/holomorphic/3328/1/t/d/3073/2"] "2-3328-13.5-c0-0-5" 1.2887545778939573 1.660888362042632 2 3328 "13.5" [] [[0.0, 0.0]] 0 true true false false 0.2032082395472503 0 1.60518017606941978150760591646 ["ModularForm/GL2/Q/holomorphic/3328/1/t/b/3073/1"] "2-3328-13.6-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "13.6" [] [[0.0, 0.0]] 0 true true false false -0.07718776385096696 0 0.949962059468041605744882648822 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/a/513/1"] "2-3328-13.6-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "13.6" [] [[0.0, 0.0]] 0 true true false false -0.07718776385096696 0 0.968232067690650475347485752375 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/b/513/1"] "2-3328-13.7-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "13.7" [] [[0.0, 0.0]] 0 true true false false -0.17077128475646636 0 0.45957740783345809860842734674 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/a/1281/1"] "2-3328-13.7-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "13.7" [] [[0.0, 0.0]] 0 true true false false -0.17077128475646636 0 1.11730136119031599691881538203 ["ModularForm/GL2/Q/holomorphic/3328/1/bl/b/1281/1"] "2-3328-13.8-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "13.8" [] [[0.0, 0.0]] 0 true true false false -0.2032082395472503 0 0.37409227542500695295941811801 ["ModularForm/GL2/Q/holomorphic/3328/1/t/a/2049/1"] "2-3328-13.8-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "13.8" [] [[0.0, 0.0]] 0 true true false false -0.2032082395472503 0 0.838829247996501694907627419700 ["ModularForm/GL2/Q/holomorphic/3328/1/t/b/2049/1"] "2-3328-13.8-c0-0-2" 1.2887545778939573 1.660888362042632 2 3328 "13.8" [] [[0.0, 0.0]] 0 true true false false -0.0782082395472503 0 0.924966298934209022985765984865 ["ModularForm/GL2/Q/holomorphic/3328/1/t/c/2049/2"] "2-3328-13.8-c0-0-3" 1.2887545778939573 1.660888362042632 2 3328 "13.8" [] [[0.0, 0.0]] 0 true true false false -0.0782082395472503 0 1.22605712779122625489598950306 ["ModularForm/GL2/Q/holomorphic/3328/1/t/d/2049/2"] "2-3328-13.8-c0-0-4" 1.2887545778939573 1.660888362042632 2 3328 "13.8" [] [[0.0, 0.0]] 0 true true false false 0.42179176045274974 0 1.63651564532410532231685125454 ["ModularForm/GL2/Q/holomorphic/3328/1/t/c/2049/1"] "2-3328-13.8-c0-0-5" 1.2887545778939573 1.660888362042632 2 3328 "13.8" [] [[0.0, 0.0]] 0 true true false false 0.42179176045274974 0 1.87572452092590075395201543127 ["ModularForm/GL2/Q/holomorphic/3328/1/t/d/2049/1"] "2-3328-208.109-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "208.109" [] [[0.0, 0.0]] 0 true true false false -0.10929176045274971 0 0.66253503940850178983487861602 ["ModularForm/GL2/Q/holomorphic/3328/1/m/b/577/1"] "2-3328-208.109-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "208.109" [] [[0.0, 0.0]] 0 true true false false 0.015708239547250296 0 1.24316633637927973032542901497 ["ModularForm/GL2/Q/holomorphic/3328/1/m/a/577/2"] "2-3328-208.109-c0-0-2" 1.2887545778939573 1.660888362042632 2 3328 "208.109" [] [[0.0, 0.0]] 0 true true false false -0.10929176045274971 0 1.40055181535485478307860847549 ["ModularForm/GL2/Q/holomorphic/3328/1/m/b/577/2"] "2-3328-208.109-c0-0-3" 1.2887545778939573 1.660888362042632 2 3328 "208.109" [] [[0.0, 0.0]] 0 true true false false -0.48429176045274974 0 2.04267720839104727382781038144 ["ModularForm/GL2/Q/holomorphic/3328/1/m/a/577/1"] "2-3328-208.125-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "208.125" [] [[0.0, 0.0]] 0 true true false false -0.2657082395472503 0 0.28089129194051545818102540862 ["ModularForm/GL2/Q/holomorphic/3328/1/r/a/2881/1"] "2-3328-208.125-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "208.125" [] [[0.0, 0.0]] 0 true true false false -0.2657082395472503 0 0.59866739518096395361422144441 ["ModularForm/GL2/Q/holomorphic/3328/1/r/a/2881/2"] "2-3328-208.125-c0-0-2" 1.2887545778939573 1.660888362042632 2 3328 "208.125" [] [[0.0, 0.0]] 0 true true false false -0.14070823954725029 0 1.28074712526457628132408735339 ["ModularForm/GL2/Q/holomorphic/3328/1/r/b/2881/2"] "2-3328-208.125-c0-0-3" 1.2887545778939573 1.660888362042632 2 3328 "208.125" [] [[0.0, 0.0]] 0 true true false false 0.35929176045274974 0 1.63916220237429244283406150341 ["ModularForm/GL2/Q/holomorphic/3328/1/r/b/2881/1"] "2-3328-208.155-c0-0-0" 1.2887545778939573 1.660888362042632 2 3328 "208.155" [] [[0.0, 0.0]] 0 true true false false -0.3125 0 0.081622336634304713557835859537 ["ModularForm/GL2/Q/holomorphic/3328/1/o/b/831/1"] "2-3328-208.155-c0-0-1" 1.2887545778939573 1.660888362042632 2 3328 "208.155" [] [[0.0, 0.0]] 0 true true false false -0.4375 0 0.47038871508759531985705999875 ["ModularForm/GL2/Q/holomorphic/3328/1/o/a/831/2"] "2-3328-208.155-c0-0-2" 1.2887545778939573 1.660888362042632 2 3328 "208.155" [] [[0.0, 0.0]] 0 true true false false 0.06250000000000001 0 0.866035675427301570078712548657 ["ModularForm/GL2/Q/holomorphic/3328/1/o/d/831/1"] "2-3328-208.155-c0-0-3" 1.2887545778939573 1.660888362042632 2 3328 "208.155" [] [[0.0, 0.0]] 0 true true false false -0.3125 0 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5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.375 0 1.78721715410286088462811938688 ["ModularForm/GL2/Q/holomorphic/3328/2/b/bd/1665/4"] "2-3328-8.5-c1-0-89" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false 0.375 0 1.82263371395638267163653410876 ["ModularForm/GL2/Q/holomorphic/3328/2/b/bd/1665/5"] "2-3328-8.5-c1-0-9" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.125 0 0.30685689583045875073179838248 ["ModularForm/GL2/Q/holomorphic/3328/2/b/a/1665/1"] "2-3328-8.5-c1-0-90" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.375 0 1.86236003989383221367791244782 ["ModularForm/GL2/Q/holomorphic/3328/2/b/v/1665/1"] "2-3328-8.5-c1-0-91" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.375 0 1.87156883686080736951059744012 ["ModularForm/GL2/Q/holomorphic/3328/2/b/y/1665/2"] "2-3328-8.5-c1-0-92" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.375 0 1.91513852812351427586914510727 ["ModularForm/GL2/Q/holomorphic/3328/2/b/bd/1665/1"] "2-3328-8.5-c1-0-93" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.375 0 2.00428988594600100214551510481 ["ModularForm/GL2/Q/holomorphic/3328/2/b/x/1665/1"] "2-3328-8.5-c1-0-94" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.125 1 2.40672729563336624583489837789 ["ModularForm/GL2/Q/holomorphic/3328/2/b/f/1665/1"] "2-3328-8.5-c1-0-95" 5.15501831157583 26.574213792682123 2 3328 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.125 0 2.56055340844846387348250556324 ["ModularForm/GL2/Q/holomorphic/3328/2/b/bb/1665/1"] # 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).