# lfunc_search downloaded from the LMFDB on 22 September 2026.
# Search link: https://www.lmfdb.org/L/2/3330/5.4
# Query "{'degree': 2, 'conductor': 3330}" returned 504 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.



"2-3330-1.1-c1-0-0"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.25204129783875144632904080893	["ModularForm/GL2/Q/holomorphic/3330/2/a/bk/1/1"]
"2-3330-1.1-c1-0-1"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.34693216058570296742577216286	["ModularForm/GL2/Q/holomorphic/3330/2/a/bj/1/1"]
"2-3330-1.1-c1-0-10"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.76184886513914383464049980410	["ModularForm/GL2/Q/holomorphic/3330/2/a/bd/1/1"]
"2-3330-1.1-c1-0-11"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.76905522154442237512605290149	["ModularForm/GL2/Q/holomorphic/3330/2/a/bk/1/2"]
"2-3330-1.1-c1-0-12"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.808697552858476107938226420465	["ModularForm/GL2/Q/holomorphic/3330/2/a/bi/1/1"]
"2-3330-1.1-c1-0-13"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.833607448087418395372710935486	["ModularForm/GL2/Q/holomorphic/3330/2/a/bj/1/2"]
"2-3330-1.1-c1-0-14"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.859662100440955635595558502502	["ModularForm/GL2/Q/holomorphic/3330/2/a/bd/1/2"]
"2-3330-1.1-c1-0-15"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.861145527082407018024994211270	["ModularForm/GL2/Q/holomorphic/3330/2/a/bh/1/3"]
"2-3330-1.1-c1-0-16"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.875384951555919833294195341928	["EllipticCurve/Q/3330/s", "ModularForm/GL2/Q/holomorphic/3330/2/a/s/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/s"]
"2-3330-1.1-c1-0-17"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.881136298997573516803859375059	["ModularForm/GL2/Q/holomorphic/3330/2/a/bl/1/1"]
"2-3330-1.1-c1-0-18"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.882877305450239840656309263477	["ModularForm/GL2/Q/holomorphic/3330/2/a/bj/1/3"]
"2-3330-1.1-c1-0-19"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.907152712227820404555887963628	["ModularForm/GL2/Q/holomorphic/3330/2/a/bi/1/2"]
"2-3330-1.1-c1-0-2"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.53403854745137626132411948490	["ModularForm/GL2/Q/holomorphic/3330/2/a/bh/1/1"]
"2-3330-1.1-c1-0-20"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.949106662803851428706954267434	["ModularForm/GL2/Q/holomorphic/3330/2/a/bh/1/2"]
"2-3330-1.1-c1-0-21"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.00195523131641969999236905200	["ModularForm/GL2/Q/holomorphic/3330/2/a/bl/1/2"]
"2-3330-1.1-c1-0-22"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.01511812341425361058763627012	["ModularForm/GL2/Q/holomorphic/3330/2/a/bf/1/1"]
"2-3330-1.1-c1-0-23"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.02149226798114432285190621341	["ModularForm/GL2/Q/holomorphic/3330/2/a/bk/1/5"]
"2-3330-1.1-c1-0-24"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.06501804660938380048832710219	["ModularForm/GL2/Q/holomorphic/3330/2/a/bc/1/2"]
"2-3330-1.1-c1-0-25"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.07428846969378408368766263038	["EllipticCurve/Q/3330/l", "ModularForm/GL2/Q/holomorphic/3330/2/a/l/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/l"]
"2-3330-1.1-c1-0-26"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.07630169819368943872459710822	["EllipticCurve/Q/3330/k", "ModularForm/GL2/Q/holomorphic/3330/2/a/k/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/k"]
"2-3330-1.1-c1-0-27"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.07643003061251412209578338863	["ModularForm/GL2/Q/holomorphic/3330/2/a/bi/1/3"]
"2-3330-1.1-c1-0-28"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.11225397641977400594990303362	["ModularForm/GL2/Q/holomorphic/3330/2/a/bf/1/2"]
"2-3330-1.1-c1-0-29"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.11392896224171524842174576504	["EllipticCurve/Q/3330/q", "ModularForm/GL2/Q/holomorphic/3330/2/a/q/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/q"]
"2-3330-1.1-c1-0-3"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.58281538244965167502065518669	["EllipticCurve/Q/3330/b", "ModularForm/GL2/Q/holomorphic/3330/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/b"]
"2-3330-1.1-c1-0-30"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.12084262714109517924855801461	["ModularForm/GL2/Q/holomorphic/3330/2/a/bl/1/3"]
"2-3330-1.1-c1-0-31"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.12328850500342707125804122399	["EllipticCurve/Q/3330/w", "ModularForm/GL2/Q/holomorphic/3330/2/a/w/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/w"]
"2-3330-1.1-c1-0-32"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.14324818618480286577693945927	["EllipticCurve/Q/3330/z", "ModularForm/GL2/Q/holomorphic/3330/2/a/z/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/z"]
"2-3330-1.1-c1-0-33"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.14418970765477512031133450528	["EllipticCurve/Q/3330/x", "ModularForm/GL2/Q/holomorphic/3330/2/a/x/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/x"]
"2-3330-1.1-c1-0-34"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.16634088288303775699053797945	["EllipticCurve/Q/3330/ba", "ModularForm/GL2/Q/holomorphic/3330/2/a/ba/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/ba"]
"2-3330-1.1-c1-0-35"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.17192536847463201195269271009	["ModularForm/GL2/Q/holomorphic/3330/2/a/bb/1/1"]
"2-3330-1.1-c1-0-36"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.17974416425608377846294685597	["ModularForm/GL2/Q/holomorphic/3330/2/a/bl/1/5"]
"2-3330-1.1-c1-0-37"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.18570928018448164103325681997	["EllipticCurve/Q/3330/r", "ModularForm/GL2/Q/holomorphic/3330/2/a/r/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/r"]
"2-3330-1.1-c1-0-38"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.20174460971488557683221711507	["EllipticCurve/Q/3330/a", "ModularForm/GL2/Q/holomorphic/3330/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/a"]
"2-3330-1.1-c1-0-39"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.21074149905000512541341956630	["EllipticCurve/Q/3330/y", "ModularForm/GL2/Q/holomorphic/3330/2/a/y/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/y"]
"2-3330-1.1-c1-0-4"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.59569632687008645347450801609	["ModularForm/GL2/Q/holomorphic/3330/2/a/bk/1/3"]
"2-3330-1.1-c1-0-40"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.25624636361396029182036603824	["EllipticCurve/Q/3330/c", "ModularForm/GL2/Q/holomorphic/3330/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/c"]
"2-3330-1.1-c1-0-41"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.33411491837942943308142579279	["EllipticCurve/Q/3330/f", "ModularForm/GL2/Q/holomorphic/3330/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/f"]
"2-3330-1.1-c1-0-42"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.34827461296543348290549238663	["ModularForm/GL2/Q/holomorphic/3330/2/a/bg/1/1"]
"2-3330-1.1-c1-0-43"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.36722901411608161833416969920	["EllipticCurve/Q/3330/g", "ModularForm/GL2/Q/holomorphic/3330/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/g"]
"2-3330-1.1-c1-0-44"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.36850503802492192231121032476	["ModularForm/GL2/Q/holomorphic/3330/2/a/bg/1/2"]
"2-3330-1.1-c1-0-45"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.39701271638754394976802518131	["ModularForm/GL2/Q/holomorphic/3330/2/a/bl/1/4"]
"2-3330-1.1-c1-0-46"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.40587043801179458996893045603	["ModularForm/GL2/Q/holomorphic/3330/2/a/bb/1/2"]
"2-3330-1.1-c1-0-47"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.42055400977912848245367112580	["EllipticCurve/Q/3330/d", "ModularForm/GL2/Q/holomorphic/3330/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/d"]
"2-3330-1.1-c1-0-48"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.42856017407325710747629050042	["EllipticCurve/Q/3330/i", "ModularForm/GL2/Q/holomorphic/3330/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/i"]
"2-3330-1.1-c1-0-49"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.45211265108040476202771386555	["EllipticCurve/Q/3330/e", "ModularForm/GL2/Q/holomorphic/3330/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/e"]
"2-3330-1.1-c1-0-5"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.64505600146968484497589973860	["ModularForm/GL2/Q/holomorphic/3330/2/a/bc/1/1"]
"2-3330-1.1-c1-0-50"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.49635501746225603083281428520	["EllipticCurve/Q/3330/j", "ModularForm/GL2/Q/holomorphic/3330/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/j"]
"2-3330-1.1-c1-0-51"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.53138459019457292030731447648	["ModularForm/GL2/Q/holomorphic/3330/2/a/bg/1/3"]
"2-3330-1.1-c1-0-52"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.59545734334537727565280593760	["EllipticCurve/Q/3330/n", "ModularForm/GL2/Q/holomorphic/3330/2/a/n/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/n"]
"2-3330-1.1-c1-0-53"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.60753780957048935469755045442	["ModularForm/GL2/Q/holomorphic/3330/2/a/be/1/1"]
"2-3330-1.1-c1-0-54"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.73957054101312727200784638700	["EllipticCurve/Q/3330/p", "ModularForm/GL2/Q/holomorphic/3330/2/a/p/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/p"]
"2-3330-1.1-c1-0-55"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.74966423477125403869326128509	["EllipticCurve/Q/3330/o", "ModularForm/GL2/Q/holomorphic/3330/2/a/o/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/o"]
"2-3330-1.1-c1-0-56"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.77239005893279657466125381004	["ModularForm/GL2/Q/holomorphic/3330/2/a/be/1/2"]
"2-3330-1.1-c1-0-57"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.79129441285692491663708007581	["EllipticCurve/Q/3330/t", "ModularForm/GL2/Q/holomorphic/3330/2/a/t/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/t"]
"2-3330-1.1-c1-0-58"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.81516757934180764073675467467	["EllipticCurve/Q/3330/u", "ModularForm/GL2/Q/holomorphic/3330/2/a/u/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/u"]
"2-3330-1.1-c1-0-59"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.82989205635176673644713429623	["EllipticCurve/Q/3330/v", "ModularForm/GL2/Q/holomorphic/3330/2/a/v/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/v"]
"2-3330-1.1-c1-0-6"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.69168407326170216395107205197	["ModularForm/GL2/Q/holomorphic/3330/2/a/bj/1/4"]
"2-3330-1.1-c1-0-7"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.71673423275031954951307857151	["EllipticCurve/Q/3330/h", "ModularForm/GL2/Q/holomorphic/3330/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/h"]
"2-3330-1.1-c1-0-8"	5.156567062793461	26.59018387308638	2	3330	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.72114804858626419919238278961	["EllipticCurve/Q/3330/m", "ModularForm/GL2/Q/holomorphic/3330/2/a/m/1/1", "ModularForm/GL2/Q/holomorphic/3330/2/a/m"]
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"2-3330-3.2-c2-0-93"	9.52554082621973	90.73592803197887	2	3330	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.3479566380076518	0	1.52681631921085424358649453393	["ModularForm/GL2/Q/holomorphic/3330/3/g/a/2591/41"]
"2-3330-3.2-c2-0-94"	9.52554082621973	90.73592803197887	2	3330	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.3479566380076518	0	1.55225522090036598251259038388	["ModularForm/GL2/Q/holomorphic/3330/3/g/a/2591/46"]
"2-3330-3.2-c2-0-95"	9.52554082621973	90.73592803197887	2	3330	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.1520433619923482	0	1.68697146546102652572485096761	["ModularForm/GL2/Q/holomorphic/3330/3/g/b/2591/5"]


# 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 <b>primitive</b> 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).