# lfunc_search downloaded from the LMFDB on 07 April 2026. # Search link: https://www.lmfdb.org/L/rational/2/355008 # Query "{'degree': 2, 'conductor': 355008, 'rational': True}" returned 114 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. "2-355008-1.1-c1-0-0" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.08794406746854903 ["EllipticCurve/Q/355008/e"] "2-355008-1.1-c1-0-1" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.1187824389520261 ["EllipticCurve/Q/355008/dt"] "2-355008-1.1-c1-0-10" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2574750209784744 ["EllipticCurve/Q/355008/bj"] "2-355008-1.1-c1-0-100" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.007731814403717 ["EllipticCurve/Q/355008/s"] "2-355008-1.1-c1-0-101" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.008005896304336 ["EllipticCurve/Q/355008/ed"] "2-355008-1.1-c1-0-102" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.020406515372356 ["EllipticCurve/Q/355008/dr"] "2-355008-1.1-c1-0-103" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 2 1.021300219221769 ["EllipticCurve/Q/355008/co"] "2-355008-1.1-c1-0-104" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.025151908261876 ["EllipticCurve/Q/355008/ej"] "2-355008-1.1-c1-0-105" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.028600014236841 ["EllipticCurve/Q/355008/dm"] "2-355008-1.1-c1-0-106" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.041110315401872 ["EllipticCurve/Q/355008/cd"] "2-355008-1.1-c1-0-107" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.067333369366463 ["EllipticCurve/Q/355008/ec"] "2-355008-1.1-c1-0-108" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.068456903897590 ["EllipticCurve/Q/355008/eb"] "2-355008-1.1-c1-0-109" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.125029718032157 ["EllipticCurve/Q/355008/ce"] "2-355008-1.1-c1-0-11" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2609562945041028 ["EllipticCurve/Q/355008/n"] "2-355008-1.1-c1-0-110" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 2 1.130699379698892 ["EllipticCurve/Q/355008/r"] "2-355008-1.1-c1-0-111" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.145008994926330 ["EllipticCurve/Q/355008/dq"] "2-355008-1.1-c1-0-112" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 2 1.146364987098993 ["EllipticCurve/Q/355008/u"] "2-355008-1.1-c1-0-113" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 2 1.287453514922307 ["EllipticCurve/Q/355008/db"] "2-355008-1.1-c1-0-12" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2619314945049921 ["EllipticCurve/Q/355008/t"] "2-355008-1.1-c1-0-13" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2639618891766309 ["EllipticCurve/Q/355008/cu"] "2-355008-1.1-c1-0-14" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2772040495115908 ["EllipticCurve/Q/355008/cs"] "2-355008-1.1-c1-0-15" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2792099163294170 ["EllipticCurve/Q/355008/bp"] "2-355008-1.1-c1-0-16" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2813872154172877 ["EllipticCurve/Q/355008/m"] "2-355008-1.1-c1-0-17" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.3022789001372546 ["EllipticCurve/Q/355008/ba"] "2-355008-1.1-c1-0-18" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.3052159988025434 ["EllipticCurve/Q/355008/bb"] "2-355008-1.1-c1-0-19" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.3533345448021691 ["EllipticCurve/Q/355008/da"] "2-355008-1.1-c1-0-2" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.1589255861306221 ["EllipticCurve/Q/355008/bx"] "2-355008-1.1-c1-0-20" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.3667141957012008 ["EllipticCurve/Q/355008/ds"] "2-355008-1.1-c1-0-21" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.3681100555730706 ["EllipticCurve/Q/355008/cl"] "2-355008-1.1-c1-0-22" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.3926080150664495 ["EllipticCurve/Q/355008/cc"] "2-355008-1.1-c1-0-23" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.3967811233620281 ["EllipticCurve/Q/355008/dh"] "2-355008-1.1-c1-0-24" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4032880473373128 ["EllipticCurve/Q/355008/i"] "2-355008-1.1-c1-0-25" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4043687585678416 ["EllipticCurve/Q/355008/bz"] "2-355008-1.1-c1-0-26" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4046390247757055 ["EllipticCurve/Q/355008/bl"] "2-355008-1.1-c1-0-27" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4099948602587379 ["EllipticCurve/Q/355008/z"] "2-355008-1.1-c1-0-28" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4186128223947586 ["EllipticCurve/Q/355008/cv"] "2-355008-1.1-c1-0-29" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4325706140715545 ["EllipticCurve/Q/355008/bn"] "2-355008-1.1-c1-0-3" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.1988625124671826 ["EllipticCurve/Q/355008/o"] "2-355008-1.1-c1-0-30" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4382777672453961 ["EllipticCurve/Q/355008/x"] "2-355008-1.1-c1-0-31" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4569352455013308 ["EllipticCurve/Q/355008/dn"] "2-355008-1.1-c1-0-32" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4587339542346577 ["EllipticCurve/Q/355008/dw"] "2-355008-1.1-c1-0-33" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4606916672536790 ["EllipticCurve/Q/355008/ei"] "2-355008-1.1-c1-0-34" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.4653208271145606 ["EllipticCurve/Q/355008/y"] "2-355008-1.1-c1-0-35" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4700022212363017 ["EllipticCurve/Q/355008/bt"] "2-355008-1.1-c1-0-36" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4729887974241681 ["EllipticCurve/Q/355008/bd"] "2-355008-1.1-c1-0-37" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4758040391270677 ["EllipticCurve/Q/355008/g"] "2-355008-1.1-c1-0-38" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.4767936937703701 ["EllipticCurve/Q/355008/h"] "2-355008-1.1-c1-0-39" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4788464351958827 ["EllipticCurve/Q/355008/cb"] "2-355008-1.1-c1-0-4" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2000227760841119 ["EllipticCurve/Q/355008/bm"] "2-355008-1.1-c1-0-40" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.4791342062906324 ["EllipticCurve/Q/355008/c"] "2-355008-1.1-c1-0-41" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4867671529032317 ["EllipticCurve/Q/355008/w"] "2-355008-1.1-c1-0-42" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4898763000430078 ["EllipticCurve/Q/355008/bv"] "2-355008-1.1-c1-0-43" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.4927419959631278 ["EllipticCurve/Q/355008/dx"] "2-355008-1.1-c1-0-44" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5163573306731809 ["EllipticCurve/Q/355008/l"] "2-355008-1.1-c1-0-45" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5205087967306154 ["EllipticCurve/Q/355008/cq"] "2-355008-1.1-c1-0-46" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5244517541017244 ["EllipticCurve/Q/355008/du"] "2-355008-1.1-c1-0-47" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5352849368802463 ["EllipticCurve/Q/355008/bs"] "2-355008-1.1-c1-0-48" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.5404239479419877 ["EllipticCurve/Q/355008/bf"] "2-355008-1.1-c1-0-49" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5589271362847522 ["EllipticCurve/Q/355008/p"] "2-355008-1.1-c1-0-5" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2179211719730827 ["EllipticCurve/Q/355008/de"] "2-355008-1.1-c1-0-50" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5630949870345712 ["EllipticCurve/Q/355008/a"] "2-355008-1.1-c1-0-51" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.5767331636527293 ["EllipticCurve/Q/355008/j"] "2-355008-1.1-c1-0-52" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.5771113135124554 ["EllipticCurve/Q/355008/b"] "2-355008-1.1-c1-0-53" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5837090379774424 ["EllipticCurve/Q/355008/cx"] "2-355008-1.1-c1-0-54" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.5844786709317604 ["EllipticCurve/Q/355008/ch"] "2-355008-1.1-c1-0-55" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.5880682625550055 ["EllipticCurve/Q/355008/do"] "2-355008-1.1-c1-0-56" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.5950454072757556 ["EllipticCurve/Q/355008/be"] "2-355008-1.1-c1-0-57" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.5965146989016291 ["EllipticCurve/Q/355008/cf"] "2-355008-1.1-c1-0-58" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.6247711534907718 ["EllipticCurve/Q/355008/ca"] "2-355008-1.1-c1-0-59" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.6303235036061953 ["EllipticCurve/Q/355008/dl"] "2-355008-1.1-c1-0-6" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2361969594932961 ["EllipticCurve/Q/355008/dc"] "2-355008-1.1-c1-0-60" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6321003570182402 ["EllipticCurve/Q/355008/cm"] "2-355008-1.1-c1-0-61" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.6324074489480314 ["EllipticCurve/Q/355008/di"] "2-355008-1.1-c1-0-62" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.6324523241240220 ["EllipticCurve/Q/355008/cw"] "2-355008-1.1-c1-0-63" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6345157311522376 ["EllipticCurve/Q/355008/d"] "2-355008-1.1-c1-0-64" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6457393579390718 ["EllipticCurve/Q/355008/bk"] "2-355008-1.1-c1-0-65" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6675145923254987 ["EllipticCurve/Q/355008/bq"] "2-355008-1.1-c1-0-66" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6697361804898539 ["EllipticCurve/Q/355008/ck"] "2-355008-1.1-c1-0-67" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6709363324642974 ["EllipticCurve/Q/355008/dd"] "2-355008-1.1-c1-0-68" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6756107683260722 ["EllipticCurve/Q/355008/cz"] "2-355008-1.1-c1-0-69" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6852114145780711 ["EllipticCurve/Q/355008/bi"] "2-355008-1.1-c1-0-7" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2475149151318296 ["EllipticCurve/Q/355008/f"] "2-355008-1.1-c1-0-70" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.6994073436848983 ["EllipticCurve/Q/355008/cp"] "2-355008-1.1-c1-0-71" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7037204587464084 ["EllipticCurve/Q/355008/bu"] "2-355008-1.1-c1-0-72" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7437862275198777 ["EllipticCurve/Q/355008/cg"] "2-355008-1.1-c1-0-73" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7492952091710322 ["EllipticCurve/Q/355008/cy"] "2-355008-1.1-c1-0-74" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7516253533876320 ["EllipticCurve/Q/355008/eg"] "2-355008-1.1-c1-0-75" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7600843747767986 ["EllipticCurve/Q/355008/bw"] "2-355008-1.1-c1-0-76" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7612812857222738 ["EllipticCurve/Q/355008/by"] "2-355008-1.1-c1-0-77" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7757916472441379 ["EllipticCurve/Q/355008/dj"] "2-355008-1.1-c1-0-78" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.7868914949150092 ["EllipticCurve/Q/355008/bc"] "2-355008-1.1-c1-0-79" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8140628678911413 ["EllipticCurve/Q/355008/q"] "2-355008-1.1-c1-0-8" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2497454087678114 ["EllipticCurve/Q/355008/bo"] "2-355008-1.1-c1-0-80" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8557258679365585 ["EllipticCurve/Q/355008/bg"] "2-355008-1.1-c1-0-81" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8599680112344507 ["EllipticCurve/Q/355008/bh"] "2-355008-1.1-c1-0-82" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8624648207031688 ["EllipticCurve/Q/355008/ct"] "2-355008-1.1-c1-0-83" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8657243022333093 ["EllipticCurve/Q/355008/dp"] "2-355008-1.1-c1-0-84" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8658685204455621 ["EllipticCurve/Q/355008/df"] "2-355008-1.1-c1-0-85" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8734145956164459 ["EllipticCurve/Q/355008/br"] "2-355008-1.1-c1-0-86" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8762003192758350 ["EllipticCurve/Q/355008/dy"] "2-355008-1.1-c1-0-87" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8845467277477050 ["EllipticCurve/Q/355008/cr"] "2-355008-1.1-c1-0-88" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.8856808677589270 ["EllipticCurve/Q/355008/ci"] "2-355008-1.1-c1-0-89" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.9052775224255893 ["EllipticCurve/Q/355008/dz"] "2-355008-1.1-c1-0-9" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.2573150238295763 ["EllipticCurve/Q/355008/eh"] "2-355008-1.1-c1-0-90" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.9279383151737649 ["EllipticCurve/Q/355008/dg"] "2-355008-1.1-c1-0-91" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.9280630262096882 ["EllipticCurve/Q/355008/ef"] "2-355008-1.1-c1-0-92" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.9350899538163576 ["EllipticCurve/Q/355008/ee"] "2-355008-1.1-c1-0-93" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.9360196019808495 ["EllipticCurve/Q/355008/cj"] "2-355008-1.1-c1-0-94" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.9543835421666629 ["EllipticCurve/Q/355008/ea"] "2-355008-1.1-c1-0-95" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 2 0.9596192184919207 ["EllipticCurve/Q/355008/k"] "2-355008-1.1-c1-0-96" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true 1 2 0.9737275251426380 ["EllipticCurve/Q/355008/cn"] "2-355008-1.1-c1-0-97" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.9894798258907607 ["EllipticCurve/Q/355008/dk"] "2-355008-1.1-c1-0-98" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 0.9951652052146338 ["EllipticCurve/Q/355008/dv"] "2-355008-1.1-c1-0-99" 53.24239994663156 2834.753152077072 2 355008 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.005995371003925 ["EllipticCurve/Q/355008/v"] # 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).