# lfunc_search downloaded from the LMFDB on 25 June 2026. # Search link: https://www.lmfdb.org/L/rational/4/768^2/1.1 # Query "{'degree': 4, 'conductor': 589824, 'rational': True}" returned 135 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, $\epsilon$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "4-768e2-1.1-c0e2-0-0" 0.6190976737980413 0.14690503763620072 4 589824 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 1.02777572148403383408704263892 ["ModularForm/GL2/Q/holomorphic/768/1/e/c", "ArtinRepresentation/2.768.8t11.c"] "4-768e2-1.1-c1e2-0-0" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.32769642696164508104003008851 ["ModularForm/GL2/Q/holomorphic/768/2/d/c"] "4-768e2-1.1-c1e2-0-1" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.53700045079540975955791842997045 ["EllipticCurve/2.0.8.1/9216.1/e", "EllipticCurve/2.0.8.1/9216.3/e", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.3/e", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.1/e"] "4-768e2-1.1-c1e2-0-10" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.73500287002208823721045648853618 ["EllipticCurve/2.0.8.1/9216.2/e", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/e", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/f", "EllipticCurve/2.0.8.1/9216.2/f"] "4-768e2-1.1-c1e2-0-11" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.77499663144732947592746935642750 ["EllipticCurve/2.0.8.1/9216.2/w", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/w", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/x", "EllipticCurve/2.0.8.1/9216.2/x"] "4-768e2-1.1-c1e2-0-12" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.77858141589193121347441626816 ["ModularForm/GL2/Q/holomorphic/768/2/d/f"] "4-768e2-1.1-c1e2-0-13" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.803024580290974129342567352760 ["EllipticCurve/2.2.12.1/4096.1/k", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-k", "ModularForm/GL2/Q/holomorphic/768/2/c/b"] "4-768e2-1.1-c1e2-0-14" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.815388463868544880605381284981602 ["EllipticCurve/2.0.8.1/9216.2/t", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/t", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/l", "EllipticCurve/2.0.8.1/9216.2/l"] "4-768e2-1.1-c1e2-0-15" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.835778173840597778927808714701 ["ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-e", "EllipticCurve/2.2.12.1/4096.1/e", "ModularForm/GL2/Q/holomorphic/768/2/c/e"] "4-768e2-1.1-c1e2-0-16" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.875626190077535716608498207062459 ["EllipticCurve/2.0.8.1/9216.2/k", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/k", "EllipticCurve/2.0.8.1/9216.2/s", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/s"] "4-768e2-1.1-c1e2-0-17" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.898833149425865420910600908462 ["ModularForm/GL2/Q/holomorphic/768/2/d/h"] "4-768e2-1.1-c1e2-0-18" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.912968318039718484977089011385 ["ModularForm/GL2/Q/holomorphic/768/2/d/e"] "4-768e2-1.1-c1e2-0-19" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.940193319874360338425704127048 ["ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-m", "EllipticCurve/2.2.12.1/4096.1/m", "EllipticCurve/2.0.8.1/9216.3/CMa", "EllipticCurve/2.0.8.1/9216.1/CMa", "ModularForm/GL2/Q/holomorphic/768/2/c/c"] "4-768e2-1.1-c1e2-0-2" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.54776244310267167881460567996559 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.1/c", "EllipticCurve/2.0.8.1/9216.1/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.3/c", "EllipticCurve/2.0.8.1/9216.3/c"] "4-768e2-1.1-c1e2-0-20" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.944281193568862877142394867146 ["EllipticCurve/2.2.12.1/4096.1/l", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-l", "ModularForm/GL2/Q/holomorphic/768/2/c/f"] "4-768e2-1.1-c1e2-0-21" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.986409188428603835741707173142616 ["EllipticCurve/2.2.12.1/4096.1/o", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-o"] "4-768e2-1.1-c1e2-0-22" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 0.986409188428603835741707173142616 ["EllipticCurve/2.0.3.1/65536.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/65536.1/a"] "4-768e2-1.1-c1e2-0-23" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.05358884779538257442246104043 ["ModularForm/GL2/Q/holomorphic/768/2/d/g"] "4-768e2-1.1-c1e2-0-24" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.06148202264849083418177125003704 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.3/a", "EllipticCurve/2.0.8.1/9216.1/a", "EllipticCurve/2.0.8.1/9216.3/a"] "4-768e2-1.1-c1e2-0-25" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.06157146329068855737319521970 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/bb", "EllipticCurve/2.0.8.1/9216.2/bb", "ModularForm/GL2/Q/holomorphic/768/2/a/l"] "4-768e2-1.1-c1e2-0-26" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.08788650049088957631470797065725 ["EllipticCurve/2.0.8.1/9216.1/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.3/b", "EllipticCurve/2.0.8.1/9216.3/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.1/b"] "4-768e2-1.1-c1e2-0-27" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.09487650279298634751041561892321 ["EllipticCurve/2.2.12.1/4096.1/c", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-c", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/65536.1/c", "EllipticCurve/2.0.3.1/65536.1/c"] "4-768e2-1.1-c1e2-0-28" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.09527349773841298009626376824493 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/b", "EllipticCurve/2.0.8.1/9216.2/b"] "4-768e2-1.1-c1e2-0-29" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.09527349773841298009626376824493 ["Genus2Curve/Q/589824/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/h", "EllipticCurve/2.0.4.1/36864.1/h"] "4-768e2-1.1-c1e2-0-3" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.59281643592350509690463923604 ["ModularForm/GL2/Q/holomorphic/768/2/d/d"] "4-768e2-1.1-c1e2-0-30" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.09582465736639484353373475455055 ["EllipticCurve/2.0.8.1/9216.2/d", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/d", "EllipticCurve/2.0.8.1/9216.2/g", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/g"] "4-768e2-1.1-c1e2-0-31" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.13947375291878398126863519213661 ["EllipticCurve/2.2.12.1/4096.1/p", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-h", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-p", "EllipticCurve/2.2.12.1/4096.1/h"] "4-768e2-1.1-c1e2-0-32" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.15979350949225944879282791156359 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/h", "EllipticCurve/2.0.8.1/9216.2/h"] "4-768e2-1.1-c1e2-0-33" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.15979350949225944879282791156359 ["EllipticCurve/2.0.4.1/36864.1/e", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/e"] "4-768e2-1.1-c1e2-0-34" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.16462166520986921322379175886 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/ba", "EllipticCurve/2.0.8.1/9216.2/ba", "ModularForm/GL2/Q/holomorphic/768/2/a/k"] "4-768e2-1.1-c1e2-0-35" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.16534733579220506723053334867150 ["EllipticCurve/2.2.12.1/4096.1/n", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-f", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-n", "EllipticCurve/2.2.12.1/4096.1/f", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.3/d", "EllipticCurve/2.0.8.1/9216.1/d", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.1/d", "EllipticCurve/2.0.8.1/9216.3/d"] "4-768e2-1.1-c1e2-0-36" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.20860922055443530129686822133117 ["EllipticCurve/2.0.8.1/9216.2/i", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/i", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/q", "EllipticCurve/2.0.8.1/9216.2/q"] "4-768e2-1.1-c1e2-0-37" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.24148568507535778914666173084920 ["EllipticCurve/2.0.8.1/9216.2/j", "EllipticCurve/2.0.8.1/9216.2/r", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/r", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/j"] "4-768e2-1.1-c1e2-0-38" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.27656804187903369079609407256772 ["EllipticCurve/2.0.4.1/36864.1/b", "EllipticCurve/2.0.4.1/36864.1/g", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/g", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/b"] "4-768e2-1.1-c1e2-0-39" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.28278081454 ["Genus2Curve/Q/589824/c"] "4-768e2-1.1-c1e2-0-4" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.60959836316426054745840675418 ["ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-a", "EllipticCurve/2.2.12.1/4096.1/a", "EllipticCurve/2.0.8.1/9216.1/CMb", "EllipticCurve/2.0.8.1/9216.3/CMb", "ModularForm/GL2/Q/holomorphic/768/2/c/d"] "4-768e2-1.1-c1e2-0-40" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.36924256495576692234243060544316 ["EllipticCurve/2.2.12.1/4096.1/j", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-j", "EllipticCurve/2.0.3.1/65536.1/d", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/65536.1/d"] "4-768e2-1.1-c1e2-0-41" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.37572229665011068043967810003498 ["EllipticCurve/2.0.8.1/9216.2/p", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/n", "EllipticCurve/2.0.8.1/9216.2/n", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/p"] "4-768e2-1.1-c1e2-0-42" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.39752451983425368394719286578451 ["ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/f", "EllipticCurve/2.0.4.1/36864.1/f"] "4-768e2-1.1-c1e2-0-43" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.39752451983425368394719286578451 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/u", "EllipticCurve/2.0.8.1/9216.2/u"] "4-768e2-1.1-c1e2-0-44" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.42267829556748512158345581078521 ["EllipticCurve/2.2.12.1/4096.1/i", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-b", "EllipticCurve/2.2.12.1/4096.1/b", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-i"] "4-768e2-1.1-c1e2-0-45" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.50163243792827999797159808653828 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.3/g", "EllipticCurve/2.0.8.1/9216.1/g", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.1/g", "EllipticCurve/2.0.8.1/9216.3/g"] "4-768e2-1.1-c1e2-0-46" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.50727956795569880796407275224861 ["EllipticCurve/2.0.8.1/9216.2/y", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/v", "EllipticCurve/2.0.8.1/9216.2/v", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/y"] "4-768e2-1.1-c1e2-0-47" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.51663712403377833666587682944114 ["EllipticCurve/2.0.8.1/9216.2/z", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/z"] "4-768e2-1.1-c1e2-0-48" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.60129609129764731842047004184279 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/65536.1/f", "EllipticCurve/2.0.3.1/65536.1/f"] "4-768e2-1.1-c1e2-0-49" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.60129609129764731842047004184279 ["EllipticCurve/2.2.12.1/4096.1/g", "ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-g"] "4-768e2-1.1-c1e2-0-5" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.66869101314413120623053151850974 ["EllipticCurve/2.0.4.1/36864.1/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/d", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/c", "EllipticCurve/2.0.4.1/36864.1/d"] "4-768e2-1.1-c1e2-0-50" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.65790914525743792045177992152 ["ModularForm/GL2/TotallyReal/2.2.12.1/holomorphic/2.2.12.1-4096.1-d", "EllipticCurve/2.2.12.1/4096.1/d", "ModularForm/GL2/Q/holomorphic/768/2/c/a"] "4-768e2-1.1-c1e2-0-51" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.73207908456882757664741684628 ["ModularForm/GL2/Q/holomorphic/768/2/d/a"] "4-768e2-1.1-c1e2-0-52" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.80842388564724944439291765894 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/a", "EllipticCurve/2.0.8.1/9216.2/a", "ModularForm/GL2/Q/holomorphic/768/2/a/i"] "4-768e2-1.1-c1e2-0-53" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.84821243380364994890921297905410 ["Genus2Curve/Q/589824/b", "EllipticCurve/2.0.4.1/36864.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/36864.1/a"] "4-768e2-1.1-c1e2-0-6" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.68078562470568089929410367530 ["EllipticCurve/2.0.8.1/9216.2/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/c", "ModularForm/GL2/Q/holomorphic/768/2/a/j"] "4-768e2-1.1-c1e2-0-7" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.69062543582573585034887958927 ["ModularForm/GL2/Q/holomorphic/768/2/d/b"] "4-768e2-1.1-c1e2-0-8" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.70350588638733060948075864561242 ["EllipticCurve/2.0.8.1/9216.2/m", "EllipticCurve/2.0.8.1/9216.2/o", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/m", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.2/o"] "4-768e2-1.1-c1e2-0-9" 2.4763906951921655 37.60768963486741 4 589824 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.71594996113128624478991539973817 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.1/f", "EllipticCurve/2.0.8.1/9216.3/f", "EllipticCurve/2.0.8.1/9216.1/f", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/9216.3/f"] "4-768e2-1.1-c2e2-0-0" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.04994528787496934983933144159 ["ModularForm/GL2/Q/holomorphic/768/3/e/c"] "4-768e2-1.1-c2e2-0-1" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.07421213470976312642838662138 ["ModularForm/GL2/Q/holomorphic/768/3/h/a"] "4-768e2-1.1-c2e2-0-2" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.19259348445552987089425141209 ["ModularForm/GL2/Q/holomorphic/768/3/g/a"] "4-768e2-1.1-c2e2-0-3" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.39015783140508021450747813285 ["ModularForm/GL2/Q/holomorphic/768/3/e/e"] "4-768e2-1.1-c2e2-0-4" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.45896411494363132745889286311 ["ModularForm/GL2/Q/holomorphic/768/3/g/b"] "4-768e2-1.1-c2e2-0-5" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.48709388338225333074117870277 ["ModularForm/GL2/Q/holomorphic/768/3/h/b"] "4-768e2-1.1-c2e2-0-6" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.50161428962335983906525389133 ["ModularForm/GL2/Q/holomorphic/768/3/e/a"] "4-768e2-1.1-c2e2-0-7" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.70652503758720576965467226410 ["ModularForm/GL2/Q/holomorphic/768/3/e/b"] "4-768e2-1.1-c2e2-0-8" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.76360069209540871973931279691 ["ModularForm/GL2/Q/holomorphic/768/3/e/d"] "4-768e2-1.1-c2e2-0-9" 4.574547442411195 437.9177452782982 4 589824 "1.1" [] [[1.0, 0.0], [1.0, 0.0]] 2 false 1 0 1.01261345116539911302159096234 ["ModularForm/GL2/Q/holomorphic/768/3/e/f"] "4-768e2-1.1-c3e2-0-0" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.18405527281707115610907958949 ["ModularForm/GL2/Q/holomorphic/768/4/c/d"] "4-768e2-1.1-c3e2-0-1" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.20809860519361954428778084452 ["ModularForm/GL2/Q/holomorphic/768/4/d/e"] "4-768e2-1.1-c3e2-0-10" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.37220561486953340513308425231 ["ModularForm/GL2/Q/holomorphic/768/4/a/f"] "4-768e2-1.1-c3e2-0-11" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.39158637021356722275932453007 ["ModularForm/GL2/Q/holomorphic/768/4/d/a"] "4-768e2-1.1-c3e2-0-12" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.39853686812500331513673711474 ["ModularForm/GL2/Q/holomorphic/768/4/c/g"] "4-768e2-1.1-c3e2-0-13" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.39940529911352392591634383942 ["ModularForm/GL2/Q/holomorphic/768/4/a/l"] "4-768e2-1.1-c3e2-0-14" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.41852356046069948294231679342 ["ModularForm/GL2/Q/holomorphic/768/4/d/c"] "4-768e2-1.1-c3e2-0-15" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.45335486353941306838209751755 ["ModularForm/GL2/Q/holomorphic/768/4/d/f"] "4-768e2-1.1-c3e2-0-16" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.52004164675105632566105046199 ["ModularForm/GL2/Q/holomorphic/768/4/c/b"] "4-768e2-1.1-c3e2-0-17" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.52789744647342078562359004105 ["ModularForm/GL2/Q/holomorphic/768/4/a/i"] "4-768e2-1.1-c3e2-0-18" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.56654290941226842393521972251 ["ModularForm/GL2/Q/holomorphic/768/4/a/h"] "4-768e2-1.1-c3e2-0-19" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.63775003804180060977512182512 ["ModularForm/GL2/Q/holomorphic/768/4/c/c"] "4-768e2-1.1-c3e2-0-2" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.21067141965087933931968724569 ["ModularForm/GL2/Q/holomorphic/768/4/d/d"] "4-768e2-1.1-c3e2-0-20" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.73308795257463845085006709329 ["ModularForm/GL2/Q/holomorphic/768/4/a/k"] "4-768e2-1.1-c3e2-0-21" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.74043864446023744859760646034 ["ModularForm/GL2/Q/holomorphic/768/4/d/o"] "4-768e2-1.1-c3e2-0-22" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.75375405850339350467830960990 ["ModularForm/GL2/Q/holomorphic/768/4/d/p"] "4-768e2-1.1-c3e2-0-23" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.78031552954546111070347212212 ["ModularForm/GL2/Q/holomorphic/768/4/d/m"] "4-768e2-1.1-c3e2-0-24" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.811579040061299962554660406386 ["ModularForm/GL2/Q/holomorphic/768/4/c/i"] "4-768e2-1.1-c3e2-0-25" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.819282415395064833124446378237 ["ModularForm/GL2/Q/holomorphic/768/4/d/k"] "4-768e2-1.1-c3e2-0-26" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.830855516769885792721987145741 ["ModularForm/GL2/Q/holomorphic/768/4/c/e"] "4-768e2-1.1-c3e2-0-27" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.871542803483639301446347941401 ["ModularForm/GL2/Q/holomorphic/768/4/c/j"] "4-768e2-1.1-c3e2-0-28" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.880728422792300369603705344654 ["ModularForm/GL2/Q/holomorphic/768/4/d/i"] "4-768e2-1.1-c3e2-0-29" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.916964568446941825513274142864 ["ModularForm/GL2/Q/holomorphic/768/4/a/o"] "4-768e2-1.1-c3e2-0-3" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.25260688879418734725232410404 ["ModularForm/GL2/Q/holomorphic/768/4/d/h"] "4-768e2-1.1-c3e2-0-30" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.11137734532814784348255038158 ["ModularForm/GL2/Q/holomorphic/768/4/a/e"] "4-768e2-1.1-c3e2-0-31" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.11509479080888699702393426189 ["ModularForm/GL2/Q/holomorphic/768/4/c/f"] "4-768e2-1.1-c3e2-0-32" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.25966626201570182391997511822 ["ModularForm/GL2/Q/holomorphic/768/4/a/g"] "4-768e2-1.1-c3e2-0-33" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.26209032781018875984168653315 ["ModularForm/GL2/Q/holomorphic/768/4/c/a"] "4-768e2-1.1-c3e2-0-34" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.47882309135144815357803456528 ["ModularForm/GL2/Q/holomorphic/768/4/d/n"] "4-768e2-1.1-c3e2-0-35" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.59139559536864167804093866819 ["ModularForm/GL2/Q/holomorphic/768/4/a/n"] "4-768e2-1.1-c3e2-0-36" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.61998869897830653345841104550 ["ModularForm/GL2/Q/holomorphic/768/4/a/p"] "4-768e2-1.1-c3e2-0-37" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 2 1.69855989237909698989824609808 ["ModularForm/GL2/Q/holomorphic/768/4/a/m"] "4-768e2-1.1-c3e2-0-4" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.29027593025749778408846499592 ["ModularForm/GL2/Q/holomorphic/768/4/d/g"] "4-768e2-1.1-c3e2-0-5" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.29535376978185990100325709068 ["ModularForm/GL2/Q/holomorphic/768/4/d/l"] "4-768e2-1.1-c3e2-0-6" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.30873367226111079688215583878 ["ModularForm/GL2/Q/holomorphic/768/4/a/j"] "4-768e2-1.1-c3e2-0-7" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.33227264278853253182183223493 ["ModularForm/GL2/Q/holomorphic/768/4/d/b"] "4-768e2-1.1-c3e2-0-8" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.35509709106441492384712583937 ["ModularForm/GL2/Q/holomorphic/768/4/c/h"] "4-768e2-1.1-c3e2-0-9" 6.731527826905926 2053.3102810844143 4 589824 "1.1" [] [[1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.36974274430274173369867956676 ["ModularForm/GL2/Q/holomorphic/768/4/d/j"] "4-768e2-1.1-c4e2-0-0" 8.91000177600389 6302.475446622494 4 589824 "1.1" [] [[2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.11038502653826517652248443430 ["ModularForm/GL2/Q/holomorphic/768/5/g/b"] "4-768e2-1.1-c4e2-0-1" 8.91000177600389 6302.475446622494 4 589824 "1.1" [] [[2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.35063829642859225894791155714 ["ModularForm/GL2/Q/holomorphic/768/5/g/a"] "4-768e2-1.1-c5e2-0-0" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.07755666173079448919690468235 ["ModularForm/GL2/Q/holomorphic/768/6/d/o"] "4-768e2-1.1-c5e2-0-1" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.091597213203832010156345804921 ["ModularForm/GL2/Q/holomorphic/768/6/d/f"] "4-768e2-1.1-c5e2-0-10" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.36920368457394916470438204888 ["ModularForm/GL2/Q/holomorphic/768/6/d/i"] "4-768e2-1.1-c5e2-0-11" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.39257299461268044998584243045 ["ModularForm/GL2/Q/holomorphic/768/6/d/l"] "4-768e2-1.1-c5e2-0-12" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.45450408718379859424597250660 ["ModularForm/GL2/Q/holomorphic/768/6/d/k"] "4-768e2-1.1-c5e2-0-13" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.45569668613869561738319675834 ["ModularForm/GL2/Q/holomorphic/768/6/d/p"] "4-768e2-1.1-c5e2-0-14" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.47198202557339614417246129567 ["ModularForm/GL2/Q/holomorphic/768/6/d/d"] "4-768e2-1.1-c5e2-0-15" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.52481609675832736763024389681 ["ModularForm/GL2/Q/holomorphic/768/6/d/n"] "4-768e2-1.1-c5e2-0-16" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.62182847707130469385101916422 ["ModularForm/GL2/Q/holomorphic/768/6/d/h"] "4-768e2-1.1-c5e2-0-17" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.62998679480631096935055434965 ["ModularForm/GL2/Q/holomorphic/768/6/a/v"] "4-768e2-1.1-c5e2-0-18" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.63233703052358116573439237965 ["ModularForm/GL2/Q/holomorphic/768/6/a/t"] "4-768e2-1.1-c5e2-0-19" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.71560099379732095173211414207 ["ModularForm/GL2/Q/holomorphic/768/6/d/r"] "4-768e2-1.1-c5e2-0-2" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.10087063399384821308900085863 ["ModularForm/GL2/Q/holomorphic/768/6/d/g"] "4-768e2-1.1-c5e2-0-20" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.74295150255320527012981752566 ["ModularForm/GL2/Q/holomorphic/768/6/d/m"] "4-768e2-1.1-c5e2-0-21" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 0.882487482806963252618206244253 ["ModularForm/GL2/Q/holomorphic/768/6/a/n"] "4-768e2-1.1-c5e2-0-22" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 1.08968808627759250045332899311 ["ModularForm/GL2/Q/holomorphic/768/6/a/p"] "4-768e2-1.1-c5e2-0-23" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 1.15838537489629053529429072418 ["ModularForm/GL2/Q/holomorphic/768/6/a/q"] "4-768e2-1.1-c5e2-0-24" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 1.20728131969562388280527066586 ["ModularForm/GL2/Q/holomorphic/768/6/a/r"] "4-768e2-1.1-c5e2-0-25" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 1.21376522113570885614009410074 ["ModularForm/GL2/Q/holomorphic/768/6/d/j"] "4-768e2-1.1-c5e2-0-26" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 1.30298389319736104847743939589 ["ModularForm/GL2/Q/holomorphic/768/6/a/s"] "4-768e2-1.1-c5e2-0-27" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 2 1.41220722220813368402747192078 ["ModularForm/GL2/Q/holomorphic/768/6/a/u"] "4-768e2-1.1-c5e2-0-3" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.11354419600341640521744503634 ["ModularForm/GL2/Q/holomorphic/768/6/d/b"] "4-768e2-1.1-c5e2-0-4" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.17034892230001688716823404247 ["ModularForm/GL2/Q/holomorphic/768/6/d/e"] "4-768e2-1.1-c5e2-0-5" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.22741334403231738324236491306 ["ModularForm/GL2/Q/holomorphic/768/6/d/q"] "4-768e2-1.1-c5e2-0-6" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.28547319233613584233877690134 ["ModularForm/GL2/Q/holomorphic/768/6/a/o"] "4-768e2-1.1-c5e2-0-7" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.30853220453386472607298916348 ["ModularForm/GL2/Q/holomorphic/768/6/d/c"] "4-768e2-1.1-c5e2-0-8" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.34030535800007220404020539403 ["ModularForm/GL2/Q/holomorphic/768/6/a/m"] "4-768e2-1.1-c5e2-0-9" 11.09841311252961 15172.024855443793 4 589824 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.36855840676720779825573101285 ["ModularForm/GL2/Q/holomorphic/768/6/d/a"] "4-768e2-1.1-c6e2-0-0" 13.292160705470183 31216.365241075284 4 589824 "1.1" [] [[3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.18944354626062671885074290404 ["ModularForm/GL2/Q/holomorphic/768/7/g/b"] "4-768e2-1.1-c6e2-0-1" 13.292160705470183 31216.365241075284 4 589824 "1.1" [] [[3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.26466731866971419917717684891 ["ModularForm/GL2/Q/holomorphic/768/7/g/a"] # 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 $1$. #$\epsilon$ (root_number) -- # The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$. The sign appears as the 4th entry in the quadruple # known as the Selberg data. #$r$ (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).