# lfunc_search downloaded from the LMFDB on 28 July 2026.
# Search link: https://www.lmfdb.org/L/2/1872/13.12/c1-0
# Query "{'degree': 2, 'conductor': 1872, 'spectral_label': 'c1-0'}" returned 490 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.



"2-1872-1.1-c1-0-0"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.38897534472433013470827993329	["EllipticCurve/Q/1872/a", "ModularForm/GL2/Q/holomorphic/1872/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/a"]
"2-1872-1.1-c1-0-1"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.58989713961815991805077071088	["EllipticCurve/Q/1872/c", "ModularForm/GL2/Q/holomorphic/1872/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/c"]
"2-1872-1.1-c1-0-10"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.06254916932353976871281040816	["EllipticCurve/Q/1872/k", "ModularForm/GL2/Q/holomorphic/1872/2/a/k/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/k"]
"2-1872-1.1-c1-0-11"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.06736363009885798558856425967	["EllipticCurve/Q/1872/p", "ModularForm/GL2/Q/holomorphic/1872/2/a/p/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/p"]
"2-1872-1.1-c1-0-12"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.07183735496432547021055620536	["EllipticCurve/Q/1872/r", "ModularForm/GL2/Q/holomorphic/1872/2/a/r/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/r"]
"2-1872-1.1-c1-0-13"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.09611714664281214369950644757	["ModularForm/GL2/Q/holomorphic/1872/2/a/x/1/2"]
"2-1872-1.1-c1-0-14"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.20578006642341497703303933331	["EllipticCurve/Q/1872/s", "ModularForm/GL2/Q/holomorphic/1872/2/a/s/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/s"]
"2-1872-1.1-c1-0-15"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.28946814831643962415194675902	["EllipticCurve/Q/1872/q", "ModularForm/GL2/Q/holomorphic/1872/2/a/q/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/q"]
"2-1872-1.1-c1-0-16"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.35061874417219692087547083371	["EllipticCurve/Q/1872/t", "ModularForm/GL2/Q/holomorphic/1872/2/a/t/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/t"]
"2-1872-1.1-c1-0-17"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.43896385530444837813510261747	["ModularForm/GL2/Q/holomorphic/1872/2/a/u/1/1"]
"2-1872-1.1-c1-0-18"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.52827597260791807015207515822	["EllipticCurve/Q/1872/d", "ModularForm/GL2/Q/holomorphic/1872/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/d"]
"2-1872-1.1-c1-0-19"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.54581469945515393401837663157	["ModularForm/GL2/Q/holomorphic/1872/2/a/y/1/2"]
"2-1872-1.1-c1-0-2"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.59372633464493589187309274913	["EllipticCurve/Q/1872/b", "ModularForm/GL2/Q/holomorphic/1872/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/b"]
"2-1872-1.1-c1-0-20"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.57297786911914117520864059523	["EllipticCurve/Q/1872/e", "ModularForm/GL2/Q/holomorphic/1872/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/e"]
"2-1872-1.1-c1-0-21"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.58843629379767240162985800185	["ModularForm/GL2/Q/holomorphic/1872/2/a/w/1/1"]
"2-1872-1.1-c1-0-22"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.61933703544595924973302121161	["EllipticCurve/Q/1872/f", "ModularForm/GL2/Q/holomorphic/1872/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/f"]
"2-1872-1.1-c1-0-23"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.63329799979128916968046932779	["ModularForm/GL2/Q/holomorphic/1872/2/a/v/1/1"]
"2-1872-1.1-c1-0-24"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.68919394139838747543180542600	["ModularForm/GL2/Q/holomorphic/1872/2/a/v/1/2"]
"2-1872-1.1-c1-0-25"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.75350296463647814243579299746	["EllipticCurve/Q/1872/n", "ModularForm/GL2/Q/holomorphic/1872/2/a/n/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/n"]
"2-1872-1.1-c1-0-26"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.77354291627119599659363661895	["EllipticCurve/Q/1872/m", "ModularForm/GL2/Q/holomorphic/1872/2/a/m/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/m"]
"2-1872-1.1-c1-0-27"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.80305900996179384247194284878	["EllipticCurve/Q/1872/o", "ModularForm/GL2/Q/holomorphic/1872/2/a/o/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/o"]
"2-1872-1.1-c1-0-28"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.87470028240941258930709038769	["ModularForm/GL2/Q/holomorphic/1872/2/a/w/1/2"]
"2-1872-1.1-c1-0-29"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.90088314593070488734462830117	["ModularForm/GL2/Q/holomorphic/1872/2/a/u/1/2"]
"2-1872-1.1-c1-0-3"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.68307113866169886014610049280	["ModularForm/GL2/Q/holomorphic/1872/2/a/y/1/1"]
"2-1872-1.1-c1-0-4"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.77347272515720828453769161406	["EllipticCurve/Q/1872/l", "ModularForm/GL2/Q/holomorphic/1872/2/a/l/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/l"]
"2-1872-1.1-c1-0-5"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.861350648696305200147674327945	["EllipticCurve/Q/1872/i", "ModularForm/GL2/Q/holomorphic/1872/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/i"]
"2-1872-1.1-c1-0-6"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.876858085797912609153868185994	["ModularForm/GL2/Q/holomorphic/1872/2/a/x/1/1"]
"2-1872-1.1-c1-0-7"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.916959205246300730998922019849	["EllipticCurve/Q/1872/g", "ModularForm/GL2/Q/holomorphic/1872/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/g"]
"2-1872-1.1-c1-0-8"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.986325425668759395181242182793	["EllipticCurve/Q/1872/j", "ModularForm/GL2/Q/holomorphic/1872/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/j"]
"2-1872-1.1-c1-0-9"	3.8662637336818726	14.947995258383695	2	1872	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.00171028893857761930540835980	["EllipticCurve/Q/1872/h", "ModularForm/GL2/Q/holomorphic/1872/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/1872/2/a/h"]
"2-1872-12.11-c1-0-0"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.48537669532568156	0	0.19529462174805934563848085285	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/7"]
"2-1872-12.11-c1-0-1"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3479566380076518	0	0.30996800475777155817225770121	["ModularForm/GL2/Q/holomorphic/1872/2/d/b/287/3"]
"2-1872-12.11-c1-0-10"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.014623304674318483	0	0.867113071570994917354808222101	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/4"]
"2-1872-12.11-c1-0-11"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.014623304674318483	0	0.871412996775347972246787546605	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/2"]
"2-1872-12.11-c1-0-12"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.014623304674318483	0	0.902619411997380848345400846040	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/5"]
"2-1872-12.11-c1-0-13"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.014623304674318483	0	1.08145217408173066190003895475	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/7"]
"2-1872-12.11-c1-0-14"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1520433619923482	0	1.11359273710884183979051351288	["ModularForm/GL2/Q/holomorphic/1872/2/d/a/287/1"]
"2-1872-12.11-c1-0-15"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.18128997134098518	0	1.22246478130947523270634487638	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/3"]
"2-1872-12.11-c1-0-16"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1520433619923482	0	1.25265986683051016485851649958	["ModularForm/GL2/Q/holomorphic/1872/2/d/a/287/3"]
"2-1872-12.11-c1-0-17"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.18128997134098518	0	1.31285692715901920997772666078	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/5"]
"2-1872-12.11-c1-0-18"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3187100286590149	0	1.42430272825997964298113751405	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/3"]
"2-1872-12.11-c1-0-19"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3479566380076518	0	1.43450590244404833217221603212	["ModularForm/GL2/Q/holomorphic/1872/2/d/b/287/2"]
"2-1872-12.11-c1-0-2"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3187100286590149	0	0.39586876152838468491542788581	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/6"]
"2-1872-12.11-c1-0-20"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3187100286590149	0	1.59259811240604206081471618644	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/1"]
"2-1872-12.11-c1-0-21"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3479566380076518	0	1.68842686424134164104966812611	["ModularForm/GL2/Q/holomorphic/1872/2/d/b/287/1"]
"2-1872-12.11-c1-0-22"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.48537669532568156	0	1.93367267665082530034396943179	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/2"]
"2-1872-12.11-c1-0-23"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.48537669532568156	0	2.02669713095022655968575859909	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/1"]
"2-1872-12.11-c1-0-3"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.48537669532568156	0	0.40111055504124501031255110695	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/8"]
"2-1872-12.11-c1-0-4"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.18128997134098518	0	0.59662774176256863606815007984	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/4"]
"2-1872-12.11-c1-0-5"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1520433619923482	0	0.59749861570805603618183730187	["ModularForm/GL2/Q/holomorphic/1872/2/d/a/287/2"]
"2-1872-12.11-c1-0-6"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3187100286590149	0	0.61473873341778295930697219192	["ModularForm/GL2/Q/holomorphic/1872/2/d/d/287/8"]
"2-1872-12.11-c1-0-7"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1520433619923482	0	0.68542269880658631296298758589	["ModularForm/GL2/Q/holomorphic/1872/2/d/a/287/4"]
"2-1872-12.11-c1-0-8"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.18128997134098518	0	0.72697283901946519589613400035	["ModularForm/GL2/Q/holomorphic/1872/2/d/c/287/6"]
"2-1872-12.11-c1-0-9"	3.8662637336818726	14.947995258383695	2	1872	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3479566380076518	0	0.73893290627576563547112468838	["ModularForm/GL2/Q/holomorphic/1872/2/d/b/287/4"]
"2-1872-13.10-c1-0-0"	3.8662637336818726	14.947995258383695	2	1872	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2663584085694929	0	0.099499888382565764563177100759	["ModularForm/GL2/Q/holomorphic/1872/2/by/o/1297/1"]
"2-1872-13.10-c1-0-1"	3.8662637336818726	14.947995258383695	2	1872	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.45731285763705	0	0.15026095004264427279420982928	["ModularForm/GL2/Q/holomorphic/1872/2/by/i/1297/2"]
"2-1872-13.10-c1-0-10"	3.8662637336818726	14.947995258383695	2	1872	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1940616711077796	0	0.75442836380928276802291193499	["ModularForm/GL2/Q/holomorphic/1872/2/by/m/1297/4"]
"2-1872-13.10-c1-0-11"	3.8662637336818726	14.947995258383695	2	1872	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.042687142362950015	0	0.75508942227275846024388682668	["ModularForm/GL2/Q/holomorphic/1872/2/by/d/1297/1"]
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"2-1872-52.47-c1-0-9"	3.8662637336818726	14.947995258383695	2	1872	"52.47"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07199740930580338	0	0.892936963348102611683054066156	["ModularForm/GL2/Q/holomorphic/1872/2/bf/o/1711/2"]


# 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).