
# lfunc_search downloaded from the LMFDB on 12 April 2026.
# Search link: https://www.lmfdb.org/L/2/42/21.2
# Query "{'degree': 2, 'conductor': 42}" returned 3697 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.



"2-42-1.1-c1-0-0"	0.5791128460752514	0.33537168848937776	2	42	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	3.62482887081886485478101246558	["EllipticCurve/Q/42/a", "ModularForm/GL2/Q/holomorphic/42/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/42/2/a/a"]
"2-42-21.17-c1-0-0"	0.5791128460752514	0.33537168848937776	2	42	"21.17"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.010086476526973061	0	3.12995661670097279809153507515	["ModularForm/GL2/Q/holomorphic/42/2/f/a/17/1"]
"2-42-21.17-c1-0-1"	0.5791128460752514	0.33537168848937776	2	42	"21.17"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.010086476526973061	0	3.42734786389487509346502907389	["ModularForm/GL2/Q/holomorphic/42/2/f/a/17/2"]
"2-42-21.20-c1-0-0"	0.5791128460752514	0.33537168848937776	2	42	"21.20"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.18668792860721248	0	2.27924580663826041386472562064	["ModularForm/GL2/Q/holomorphic/42/2/d/a/41/3"]
"2-42-21.20-c1-0-1"	0.5791128460752514	0.33537168848937776	2	42	"21.20"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.06331207139278754	0	3.38201100167339061682663398425	["ModularForm/GL2/Q/holomorphic/42/2/d/a/41/4"]
"2-42-21.20-c1-0-2"	0.5791128460752514	0.33537168848937776	2	42	"21.20"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.06331207139278754	0	3.46085368216767048570007564189	["ModularForm/GL2/Q/holomorphic/42/2/d/a/41/2"]
"2-42-21.20-c1-0-3"	0.5791128460752514	0.33537168848937776	2	42	"21.20"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.18668792860721248	0	4.30648197427357298034382340907	["ModularForm/GL2/Q/holomorphic/42/2/d/a/41/1"]
"2-42-21.5-c1-0-0"	0.5791128460752514	0.33537168848937776	2	42	"21.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.010086476526973061	0	2.70122917254672330594561681224	["ModularForm/GL2/Q/holomorphic/42/2/f/a/5/1"]
"2-42-21.5-c1-0-1"	0.5791128460752514	0.33537168848937776	2	42	"21.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.010086476526973061	0	3.70646706726554967046764789054	["ModularForm/GL2/Q/holomorphic/42/2/f/a/5/2"]
"2-42-7.2-c1-0-0"	0.5791128460752514	0.33537168848937776	2	42	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.14649371361272054	0	2.46289452936338301129272717961	["ModularForm/GL2/Q/holomorphic/42/2/e/a/37/1"]
"2-42-7.2-c1-0-1"	0.5791128460752514	0.33537168848937776	2	42	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.12632076055877442	0	4.39806420206531623293129256174	["ModularForm/GL2/Q/holomorphic/42/2/e/b/37/1"]
"2-42-7.4-c1-0-0"	0.5791128460752514	0.33537168848937776	2	42	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.12632076055877442	0	2.83236694053324174912444359948	["ModularForm/GL2/Q/holomorphic/42/2/e/b/25/1"]
"2-42-7.4-c1-0-1"	0.5791128460752514	0.33537168848937776	2	42	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.14649371361272054	0	3.82388470647963604033512192263	["ModularForm/GL2/Q/holomorphic/42/2/e/a/25/1"]
"2-42-21.11-c2-0-0"	1.0697743268153557	1.1444171103132472	2	42	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.4050319016762627	0	0.67547929323321649937980912680	["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/1"]
"2-42-21.11-c2-0-1"	1.0697743268153557	1.1444171103132472	2	42	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.04554853592115429	0	2.36855596773987885080361239393	["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/2"]
"2-42-21.11-c2-0-2"	1.0697743268153557	1.1444171103132472	2	42	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.0077304283528976045	0	2.66180964612968656785158117705	["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/4"]
"2-42-21.11-c2-0-3"	1.0697743268153557	1.1444171103132472	2	42	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.14750653384500276	0	2.80747555164711503271266000506	["ModularForm/GL2/Q/holomorphic/42/3/h/a/11/1"]
"2-42-21.11-c2-0-4"	1.0697743268153557	1.1444171103132472	2	42	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.03933308587561003	0	3.24154603880256607681780776282	["ModularForm/GL2/Q/holomorphic/42/3/h/a/11/2"]
"2-42-21.11-c2-0-5"	1.0697743268153557	1.1444171103132472	2	42	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.16532343872487346	0	3.83322646708001216848073302714	["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/3"]
"2-42-21.2-c2-0-0"	1.0697743268153557	1.1444171103132472	2	42	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.14750653384500276	0	1.39112087438542918662457635971	["ModularForm/GL2/Q/holomorphic/42/3/h/a/23/1"]
"2-42-21.2-c2-0-1"	1.0697743268153557	1.1444171103132472	2	42	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.16532343872487346	0	2.12843528403547952178587957365	["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/3"]
"2-42-21.2-c2-0-2"	1.0697743268153557	1.1444171103132472	2	42	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.04554853592115429	0	2.47624137765638921638372235460	["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/2"]
"2-42-21.2-c2-0-3"	1.0697743268153557	1.1444171103132472	2	42	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.0077304283528976045	0	3.01201525965927102144182932257	["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/4"]
"2-42-21.2-c2-0-4"	1.0697743268153557	1.1444171103132472	2	42	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.03933308587561003	0	3.21653654134617585231728888020	["ModularForm/GL2/Q/holomorphic/42/3/h/a/23/2"]
"2-42-21.2-c2-0-5"	1.0697743268153557	1.1444171103132472	2	42	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.4050319016762627	0	3.91588356256673611795640862293	["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/1"]
"2-42-3.2-c2-0-0"	1.0697743268153557	1.1444171103132472	2	42	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.32812640472793253	0	1.23197583297330044693123404601	["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/3"]
"2-42-3.2-c2-0-1"	1.0697743268153557	1.1444171103132472	2	42	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.1718735952720675	0	2.25442640976318514445653685193	["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/4"]
"2-42-3.2-c2-0-2"	1.0697743268153557	1.1444171103132472	2	42	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.1718735952720675	0	3.44301175405416403260060461427	["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/2"]
"2-42-3.2-c2-0-3"	1.0697743268153557	1.1444171103132472	2	42	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.32812640472793253	0	4.00088304407135574630311318648	["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/1"]
"2-42-7.3-c2-0-0"	1.0697743268153557	1.1444171103132472	2	42	"7.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.14679709583952122	0	2.36411800528140250691883631837	["ModularForm/GL2/Q/holomorphic/42/3/g/a/31/2"]
"2-42-7.3-c2-0-1"	1.0697743268153557	1.1444171103132472	2	42	"7.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.10645118973162895	0	2.72031556201279849766172301524	["ModularForm/GL2/Q/holomorphic/42/3/g/a/31/1"]
"2-42-7.5-c2-0-0"	1.0697743268153557	1.1444171103132472	2	42	"7.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.10645118973162895	0	2.20604359483068962765831611063	["ModularForm/GL2/Q/holomorphic/42/3/g/a/19/1"]
"2-42-7.5-c2-0-1"	1.0697743268153557	1.1444171103132472	2	42	"7.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.14679709583952122	0	3.52432439487018561702569581882	["ModularForm/GL2/Q/holomorphic/42/3/g/a/19/2"]
"2-42-7.6-c2-0-0"	1.0697743268153557	1.1444171103132472	2	42	"7.6"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.175280691521465	0	1.47387573767770102315147410442	["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/2"]
"2-42-7.6-c2-0-1"	1.0697743268153557	1.1444171103132472	2	42	"7.6"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.05190483430704006	0	2.65424106596318642352946289616	["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/4"]
"2-42-7.6-c2-0-2"	1.0697743268153557	1.1444171103132472	2	42	"7.6"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.175280691521465	0	2.93119924689143621118693881153	["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/1"]
"2-42-7.6-c2-0-3"	1.0697743268153557	1.1444171103132472	2	42	"7.6"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.05190483430704006	0	3.31428425515721586056396944495	["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/3"]
"2-42-1.1-c3-0-0"	1.574191926113556	2.478080220241107	2	42	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.0	0	2.31016872062034107209262790600	["ModularForm/GL2/Q/holomorphic/42/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/42/4/a/a"]
"2-42-1.1-c3-0-1"	1.574191926113556	2.478080220241107	2	42	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.0	0	2.60588720017564028889637504318	["ModularForm/GL2/Q/holomorphic/42/4/a/b/1/1", "ModularForm/GL2/Q/holomorphic/42/4/a/b"]
"2-42-21.17-c3-0-0"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.14374757875432342	0	1.21442106286669964666587283670	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/2"]
"2-42-21.17-c3-0-1"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.27368251401058724	0	1.23784660790389368779327579723	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/3"]
"2-42-21.17-c3-0-2"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.06654712002222955	0	1.98082379158613252076274990060	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/6"]
"2-42-21.17-c3-0-3"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.09794607534271514	0	2.49783835648320092326696177864	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/4"]
"2-42-21.17-c3-0-4"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.3221255385397444	0	2.58873764215124798777500597205	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/1"]
"2-42-21.17-c3-0-5"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.0757309622693895	0	2.91819872575656085997738859166	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/8"]
"2-42-21.17-c3-0-6"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.19155711439681347	0	3.28149325960991510892352155582	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/7"]
"2-42-21.17-c3-0-7"	1.574191926113556	2.478080220241107	2	42	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.30190056447357544	0	3.55601823750063915409284878848	["ModularForm/GL2/Q/holomorphic/42/4/f/a/17/5"]
"2-42-21.20-c3-0-0"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4083158956420014	0	0.06986784575391022400258742491	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/1"]
"2-42-21.20-c3-0-1"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.43395606122796576	0	0.846162881185845017571493382553	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/6"]
"2-42-21.20-c3-0-2"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.2190627076611827	0	1.81409075161617370822042647845	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/8"]
"2-42-21.20-c3-0-3"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.005297126752852992	0	1.89935550184982815033040953295	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/3"]
"2-42-21.20-c3-0-4"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.005297126752852992	0	2.32043329302651696119709591756	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/7"]
"2-42-21.20-c3-0-5"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.2190627076611827	0	3.01519619078416127378215616457	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/4"]
"2-42-21.20-c3-0-6"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.4083158956420014	0	3.48510342266078607840559528230	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/5"]
"2-42-21.20-c3-0-7"	1.574191926113556	2.478080220241107	2	42	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.43395606122796576	0	4.06735620945415211898759578671	["ModularForm/GL2/Q/holomorphic/42/4/d/a/41/2"]
"2-42-21.5-c3-0-0"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.3221255385397444	0	0.28038133432348573849515631207	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/1"]
"2-42-21.5-c3-0-1"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.30190056447357544	0	1.10176397680589148327476835046	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/5"]
"2-42-21.5-c3-0-2"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.09794607534271514	0	1.50511377620774824331762936665	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/4"]
"2-42-21.5-c3-0-3"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.19155711439681347	0	1.85197233113115655136756724682	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/7"]
"2-42-21.5-c3-0-4"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.14374757875432342	0	2.00242272473797244360681682431	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/2"]
"2-42-21.5-c3-0-5"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.0757309622693895	0	2.45680303265975874847664943562	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/8"]
"2-42-21.5-c3-0-6"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.06654712002222955	0	2.84514711539733920434908448416	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/6"]
"2-42-21.5-c3-0-7"	1.574191926113556	2.478080220241107	2	42	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.27368251401058724	0	2.88586236306103118510840540375	["ModularForm/GL2/Q/holomorphic/42/4/f/a/5/3"]
"2-42-7.2-c3-0-0"	1.574191926113556	2.478080220241107	2	42	"7.2"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.449153750081285	0	0.41850663403096990490243996388	["ModularForm/GL2/Q/holomorphic/42/4/e/c/37/1"]
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"2-42-7.4-c29-0-25"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.026827089619396765	0	0.889518753488593545247526363081	["ModularForm/GL2/Q/holomorphic/42/30/e/d/25/10"]
"2-42-7.4-c29-0-26"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.21492522027277242	0	0.892257913810836390794231377677	["ModularForm/GL2/Q/holomorphic/42/30/e/a/25/7"]
"2-42-7.4-c29-0-27"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.13468529351165048	0	0.931225017842590559202944336039	["ModularForm/GL2/Q/holomorphic/42/30/e/d/25/7"]
"2-42-7.4-c29-0-28"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.44868496462374846	0	0.973623599762826988923616923687	["ModularForm/GL2/Q/holomorphic/42/30/e/c/25/7"]
"2-42-7.4-c29-0-29"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.48523651555843417	0	0.977982334046304146790647176860	["ModularForm/GL2/Q/holomorphic/42/30/e/c/25/6"]
"2-42-7.4-c29-0-3"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.24614563240640977	0	0.19117552893851345809687184446	["ModularForm/GL2/Q/holomorphic/42/30/e/c/25/10"]
"2-42-7.4-c29-0-30"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.45898435265857507	0	1.15316095148303518346749027290	["ModularForm/GL2/Q/holomorphic/42/30/e/a/25/4"]
"2-42-7.4-c29-0-31"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.3979637525736331	0	1.17271405995046568662640932940	["ModularForm/GL2/Q/holomorphic/42/30/e/d/25/5"]
"2-42-7.4-c29-0-32"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.42101940261294485	0	1.18470606760952879929122754749	["ModularForm/GL2/Q/holomorphic/42/30/e/b/25/3"]
"2-42-7.4-c29-0-33"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.2433629442961408	0	1.19218695828336050648600116499	["ModularForm/GL2/Q/holomorphic/42/30/e/b/25/9"]
"2-42-7.4-c29-0-34"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.25947730948916053	0	1.24093883408368993144170810585	["ModularForm/GL2/Q/holomorphic/42/30/e/a/25/2"]
"2-42-7.4-c29-0-35"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.2595757860163068	0	1.27068655671111603003215496253	["ModularForm/GL2/Q/holomorphic/42/30/e/b/25/4"]
"2-42-7.4-c29-0-36"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.22413958377692858	0	1.53213936418931134508711415283	["ModularForm/GL2/Q/holomorphic/42/30/e/c/25/3"]
"2-42-7.4-c29-0-37"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.1789915068167551	0	1.54550045003981460433523956085	["ModularForm/GL2/Q/holomorphic/42/30/e/c/25/4"]
"2-42-7.4-c29-0-4"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.19734470470643908	0	0.19303007117585686016013571565	["ModularForm/GL2/Q/holomorphic/42/30/e/d/25/8"]
"2-42-7.4-c29-0-5"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.38723423339890034	0	0.29014973232364958626794425000	["ModularForm/GL2/Q/holomorphic/42/30/e/a/25/8"]
"2-42-7.4-c29-0-6"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.11212723650167453	0	0.29595751927578084525782305077	["ModularForm/GL2/Q/holomorphic/42/30/e/a/25/3"]
"2-42-7.4-c29-0-7"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.4438017578611654	0	0.31379465141388919111484337405	["ModularForm/GL2/Q/holomorphic/42/30/e/b/25/7"]
"2-42-7.4-c29-0-8"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	-0.4266429381560727	0	0.31714319541310476537949774767	["ModularForm/GL2/Q/holomorphic/42/30/e/d/25/9"]
"2-42-7.4-c29-0-9"	14.958869543141953	223.76777800873995	2	42	"7.4"	[]	[[14.5, 0.0]]	29	true	true	false	false	0.11619577510086765	0	0.34147318599228280509999311039	["ModularForm/GL2/Q/holomorphic/42/30/e/c/25/2"]


# 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).


