# lfunc_search downloaded from the LMFDB on 14 June 2026. # Search link: https://www.lmfdb.org/L/2/1700/85.64/c1-0 # Query "{'degree': 2, 'conductor': 1700, 'spectral_label': 'c1-0'}" returned 218 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. "2-1700-1.1-c1-0-0" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.26961809085309071036914408555 ["ModularForm/GL2/Q/holomorphic/1700/2/a/h/1/2"] "2-1700-1.1-c1-0-1" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.53961782848774431033963339417 ["ModularForm/GL2/Q/holomorphic/1700/2/a/e/1/1"] "2-1700-1.1-c1-0-10" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.06691284432912638613285397630 ["ModularForm/GL2/Q/holomorphic/1700/2/a/h/1/4"] "2-1700-1.1-c1-0-11" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.09885205471654667388017405349 ["EllipticCurve/Q/1700/b", "ModularForm/GL2/Q/holomorphic/1700/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/1700/2/a/b"] "2-1700-1.1-c1-0-12" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.19340597475152719128733009340 ["ModularForm/GL2/Q/holomorphic/1700/2/a/g/1/3"] "2-1700-1.1-c1-0-13" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.27335207607444955251057342926 ["ModularForm/GL2/Q/holomorphic/1700/2/a/i/1/5"] "2-1700-1.1-c1-0-14" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.30367206255438756102986986706 ["ModularForm/GL2/Q/holomorphic/1700/2/a/g/1/4"] "2-1700-1.1-c1-0-15" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.34754957926582200184145435931 ["ModularForm/GL2/Q/holomorphic/1700/2/a/f/1/1"] "2-1700-1.1-c1-0-16" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.39878557995822729655014583546 ["ModularForm/GL2/Q/holomorphic/1700/2/a/d/1/1"] "2-1700-1.1-c1-0-17" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.42202089570582602564211875231 ["ModularForm/GL2/Q/holomorphic/1700/2/a/i/1/4"] "2-1700-1.1-c1-0-18" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.45147819961693176534262986127 ["ModularForm/GL2/Q/holomorphic/1700/2/a/f/1/2"] "2-1700-1.1-c1-0-19" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.45457895523075657036915804987 ["ModularForm/GL2/Q/holomorphic/1700/2/a/e/1/3"] "2-1700-1.1-c1-0-2" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.58380063245629289529229000801 ["ModularForm/GL2/Q/holomorphic/1700/2/a/i/1/2"] "2-1700-1.1-c1-0-20" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.55979449114536678576992400339 ["ModularForm/GL2/Q/holomorphic/1700/2/a/h/1/5"] "2-1700-1.1-c1-0-21" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.59165464203817659633355943598 ["EllipticCurve/Q/1700/a", "ModularForm/GL2/Q/holomorphic/1700/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/1700/2/a/a"] "2-1700-1.1-c1-0-22" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.80479644723143014792244333759 ["ModularForm/GL2/Q/holomorphic/1700/2/a/d/1/2"] "2-1700-1.1-c1-0-23" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.89813994928174575663005614653 ["EllipticCurve/Q/1700/c", "ModularForm/GL2/Q/holomorphic/1700/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/1700/2/a/c"] "2-1700-1.1-c1-0-24" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.99283240020093811944547368743 ["ModularForm/GL2/Q/holomorphic/1700/2/a/f/1/3"] "2-1700-1.1-c1-0-25" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 2.00557136759948378968906332074 ["ModularForm/GL2/Q/holomorphic/1700/2/a/f/1/4"] "2-1700-1.1-c1-0-3" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.64063997081995567349874009823 ["ModularForm/GL2/Q/holomorphic/1700/2/a/i/1/1"] "2-1700-1.1-c1-0-4" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.65834964393018390452831069748 ["ModularForm/GL2/Q/holomorphic/1700/2/a/g/1/2"] "2-1700-1.1-c1-0-5" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.813224148095549335930296401439 ["ModularForm/GL2/Q/holomorphic/1700/2/a/g/1/1"] "2-1700-1.1-c1-0-6" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.826616203711609529673443946636 ["ModularForm/GL2/Q/holomorphic/1700/2/a/e/1/2"] "2-1700-1.1-c1-0-7" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.862273208676391869168522726897 ["ModularForm/GL2/Q/holomorphic/1700/2/a/h/1/3"] "2-1700-1.1-c1-0-8" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.989619336458783456183100046123 ["ModularForm/GL2/Q/holomorphic/1700/2/a/h/1/1"] "2-1700-1.1-c1-0-9" 3.6843681064217337 13.574568343617672 2 1700 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.03349638331192142224068342010 ["ModularForm/GL2/Q/holomorphic/1700/2/a/i/1/3"] "2-1700-17.13-c1-0-0" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.43534797111584267 0 0.14622968017474040235659370633 ["ModularForm/GL2/Q/holomorphic/1700/2/o/d/1101/4"] "2-1700-17.13-c1-0-1" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.49722219476980845 0 0.23113388085613804431567867884 ["ModularForm/GL2/Q/holomorphic/1700/2/o/g/1101/5"] "2-1700-17.13-c1-0-10" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.14449478259434237 0 0.73305218182036777496926018034 ["ModularForm/GL2/Q/holomorphic/1700/2/o/a/1101/1"] "2-1700-17.13-c1-0-11" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.3036165779797826 0 0.823195045996668458118471131676 ["ModularForm/GL2/Q/holomorphic/1700/2/o/e/1101/6"] "2-1700-17.13-c1-0-12" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.061356030910793 0 1.03114020403517521804244964905 ["ModularForm/GL2/Q/holomorphic/1700/2/o/f/1101/5"] "2-1700-17.13-c1-0-13" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.18610838580732858 0 1.12217667386095782879300318619 ["ModularForm/GL2/Q/holomorphic/1700/2/o/c/1101/1"] "2-1700-17.13-c1-0-14" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.4524286163559021 0 1.23934056823243277886264837729 ["ModularForm/GL2/Q/holomorphic/1700/2/o/f/1101/1"] "2-1700-17.13-c1-0-15" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.00833223601983222 0 1.25000829069410014214822211963 ["ModularForm/GL2/Q/holomorphic/1700/2/o/d/1101/6"] "2-1700-17.13-c1-0-16" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.24653881473759123 0 1.26919645371083063566680387669 ["ModularForm/GL2/Q/holomorphic/1700/2/o/g/1101/3"] "2-1700-17.13-c1-0-17" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.2660554071225093 0 1.35068875910982546930723257200 ["ModularForm/GL2/Q/holomorphic/1700/2/o/g/1101/2"] "2-1700-17.13-c1-0-18" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.04757138364409789 0 1.41017661832646525079090456496 ["ModularForm/GL2/Q/holomorphic/1700/2/o/f/1101/6"] "2-1700-17.13-c1-0-19" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.08577127747894142 0 1.41213292102160186999607250667 ["ModularForm/GL2/Q/holomorphic/1700/2/o/g/1101/6"] "2-1700-17.13-c1-0-2" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.28556736803945143 0 0.31254602560211749816412973651 ["ModularForm/GL2/Q/holomorphic/1700/2/o/f/1101/3"] "2-1700-17.13-c1-0-20" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.49722219476980845 0 1.43923128168494971252232475200 ["ModularForm/GL2/Q/holomorphic/1700/2/o/e/1101/2"] "2-1700-17.13-c1-0-21" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.07710225165106913 0 1.46281102434777747700825596046 ["ModularForm/GL2/Q/holomorphic/1700/2/o/d/1101/5"] "2-1700-17.13-c1-0-22" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.21443263196054857 0 1.48552101533608723693368629354 ["ModularForm/GL2/Q/holomorphic/1700/2/o/f/1101/4"] "2-1700-17.13-c1-0-23" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.2660554071225093 0 1.51624763162368892231872885964 ["ModularForm/GL2/Q/holomorphic/1700/2/o/e/1101/5"] "2-1700-17.13-c1-0-24" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.43864396908920705 0 1.73535045158698843079792193738 ["ModularForm/GL2/Q/holomorphic/1700/2/o/f/1101/2"] "2-1700-17.13-c1-0-25" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.24653881473759123 0 1.74316678111201648940827237082 ["ModularForm/GL2/Q/holomorphic/1700/2/o/e/1101/4"] "2-1700-17.13-c1-0-26" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.3036165779797826 0 1.86718861077549264095937575319 ["ModularForm/GL2/Q/holomorphic/1700/2/o/g/1101/1"] "2-1700-17.13-c1-0-27" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.39495093880262006 0 1.96485277136676208173940267379 ["ModularForm/GL2/Q/holomorphic/1700/2/o/d/1101/2"] "2-1700-17.13-c1-0-3" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.08577127747894142 0 0.37211035407486120430639982277 ["ModularForm/GL2/Q/holomorphic/1700/2/o/e/1101/1"] "2-1700-17.13-c1-0-4" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.15797279874933742 0 0.42208496978419811914774910312 ["ModularForm/GL2/Q/holomorphic/1700/2/o/d/1101/3"] "2-1700-17.13-c1-0-5" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.1649737368202528 0 0.51929028132355137298422545202 ["ModularForm/GL2/Q/holomorphic/1700/2/o/e/1101/3"] "2-1700-17.13-c1-0-6" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.4750979509960132 0 0.54197180669556903808739161815 ["ModularForm/GL2/Q/holomorphic/1700/2/o/c/1101/2"] "2-1700-17.13-c1-0-7" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.20849125822524792 0 0.63803692906692243828270986910 ["ModularForm/GL2/Q/holomorphic/1700/2/o/d/1101/1"] "2-1700-17.13-c1-0-8" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.1649737368202528 0 0.66436748084986328574776673754 ["ModularForm/GL2/Q/holomorphic/1700/2/o/g/1101/4"] "2-1700-17.13-c1-0-9" 3.6843681064217337 13.574568343617672 2 1700 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.35550521740565766 0 0.68859354529205942312562696170 ["ModularForm/GL2/Q/holomorphic/1700/2/o/b/1101/1"] "2-1700-17.16-c1-0-0" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.25295340687344076 0 0.07743718183433679975763670456 ["ModularForm/GL2/Q/holomorphic/1700/2/c/c/101/1"] "2-1700-17.16-c1-0-1" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.15706576944956221 0 0.17980801271196491515942006149 ["ModularForm/GL2/Q/holomorphic/1700/2/c/b/101/1"] "2-1700-17.16-c1-0-10" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.09774885323061938 0 0.872073984171068249215481975416 ["ModularForm/GL2/Q/holomorphic/1700/2/c/e/101/2"] "2-1700-17.16-c1-0-11" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.10963313754927129 0 0.894032806924211600446518534041 ["ModularForm/GL2/Q/holomorphic/1700/2/c/c/101/4"] "2-1700-17.16-c1-0-12" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.023055246592774158 0 0.976174775125654133513571378987 ["ModularForm/GL2/Q/holomorphic/1700/2/c/e/101/4"] "2-1700-17.16-c1-0-13" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.38734575413049926 0 0.999282315324182723877604650986 ["ModularForm/GL2/Q/holomorphic/1700/2/c/b/101/5"] "2-1700-17.16-c1-0-14" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.09774885323061938 0 1.03360338415207791091580624804 ["ModularForm/GL2/Q/holomorphic/1700/2/c/e/101/7"] "2-1700-17.16-c1-0-15" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.25295340687344076 0 1.12786965950377471893797090658 ["ModularForm/GL2/Q/holomorphic/1700/2/c/d/101/1"] "2-1700-17.16-c1-0-16" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.023055246592774158 0 1.14306694283178014155544142680 ["ModularForm/GL2/Q/holomorphic/1700/2/c/e/101/5"] "2-1700-17.16-c1-0-17" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.10963313754927129 0 1.15208866719593634262266204960 ["ModularForm/GL2/Q/holomorphic/1700/2/c/c/101/3"] "2-1700-17.16-c1-0-18" 3.6843681064217337 13.574568343617672 2 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["ModularForm/GL2/Q/holomorphic/1700/2/c/c/101/6"] "2-1700-17.16-c1-0-22" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.15706576944956221 0 1.50526787594996120331153851388 ["ModularForm/GL2/Q/holomorphic/1700/2/c/b/101/6"] "2-1700-17.16-c1-0-23" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.4022511467693807 0 1.52552274755825237788277716957 ["ModularForm/GL2/Q/holomorphic/1700/2/c/e/101/1"] "2-1700-17.16-c1-0-24" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.38734575413049926 0 1.67230702990943102546744310150 ["ModularForm/GL2/Q/holomorphic/1700/2/c/b/101/2"] "2-1700-17.16-c1-0-25" 3.6843681064217337 13.574568343617672 2 1700 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.48568961224215074 0 1.82170386992166948031036914120 ["ModularForm/GL2/Q/holomorphic/1700/2/c/c/101/2"] "2-1700-17.16-c1-0-26" 3.6843681064217337 13.574568343617672 2 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0.38842981316275308781295632052 ["ModularForm/GL2/Q/holomorphic/1700/2/g/c/849/4"] "2-1700-85.84-c1-0-5" 3.6843681064217337 13.574568343617672 2 1700 "85.84" [] [[0.5, 0.0]] 1 true true false false -0.3135539453052827 0 0.41951840691287201519515087286 ["ModularForm/GL2/Q/holomorphic/1700/2/g/c/849/2"] "2-1700-85.84-c1-0-6" 3.6843681064217337 13.574568343617672 2 1700 "85.84" [] [[0.5, 0.0]] 1 true true false false -0.21415867127594537 0 0.45045287123403588630305910448 ["ModularForm/GL2/Q/holomorphic/1700/2/g/d/849/5"] "2-1700-85.84-c1-0-7" 3.6843681064217337 13.574568343617672 2 1700 "85.84" [] [[0.5, 0.0]] 1 true true false false -0.23085757827477887 0 0.46193419173006994200804632407 ["ModularForm/GL2/Q/holomorphic/1700/2/g/b/849/2"] "2-1700-85.84-c1-0-8" 3.6843681064217337 13.574568343617672 2 1700 "85.84" [] [[0.5, 0.0]] 1 true true false false -0.1816301495661602 0 0.46600647250639356484340908069 ["ModularForm/GL2/Q/holomorphic/1700/2/g/b/849/4"] "2-1700-85.84-c1-0-9" 3.6843681064217337 13.574568343617672 2 1700 "85.84" [] [[0.5, 0.0]] 1 true true false false -0.16189780341693408 0 0.59144421981025860017666748579 ["ModularForm/GL2/Q/holomorphic/1700/2/g/d/849/3"] # 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).