# lfunc_search downloaded from the LMFDB on 27 June 2026. # Search link: https://www.lmfdb.org/L/2/2^7/128.109/c1-0 # Query "{'degree': 2, 'conductor': 128, 'spectral_label': 'c1-0'}" returned 264 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. "2-2e7-1.1-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.73703685485230483844290438322 ["EllipticCurve/Q/128/b", "ModularForm/GL2/Q/holomorphic/128/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/128/2/a/b"] "2-2e7-1.1-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 2.30676446591740616355718333113 ["EllipticCurve/Q/128/c", "ModularForm/GL2/Q/holomorphic/128/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/128/2/a/c"] "2-2e7-1.1-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 2.46608091328452001690059553408 ["EllipticCurve/Q/128/d", "ModularForm/GL2/Q/holomorphic/128/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/128/2/a/d"] "2-2e7-1.1-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 3.36730269661969347724618284285 ["EllipticCurve/Q/128/a", "ModularForm/GL2/Q/holomorphic/128/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/128/2/a/a"] "2-2e7-128.101-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false -0.42281830982233404 0 0.11664435912694360078336561450 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/3"] "2-2e7-128.101-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false -0.21092002888136077 0 0.60796130462897932007799359271 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/9"] "2-2e7-128.101-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.030422108194951548 0 2.54110431857810722440460355564 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/6"] "2-2e7-128.101-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false -0.0412209251198049 0 2.55360563479846038432935825269 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/15"] "2-2e7-128.101-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.11538900917735488 0 3.15063414276425626669239177545 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/14"] "2-2e7-128.101-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.3224080031866884 0 3.29517762567698438732748443267 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/7"] "2-2e7-128.101-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.3753333185668557 0 3.35730293431655633083154830900 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/11"] "2-2e7-128.101-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false -0.37416745284323655 0 1.11476966572450945852966219670 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/12"] "2-2e7-128.101-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false -0.3140111651650164 0 1.25542552225988312749750955722 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/8"] "2-2e7-128.101-c1-0-4" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false -0.06804628103861464 0 1.47461184042137867245559421968 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/2"] "2-2e7-128.101-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.09375472367893647 0 1.84704764797847508453378414643 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/5"] "2-2e7-128.101-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.19535261716066935 0 1.90451974591260928895226234331 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/4"] "2-2e7-128.101-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.02048013331729377 0 2.05313092954977797160976258706 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/13"] "2-2e7-128.101-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false 0.03486794831387463 0 2.23308091454899859426504078474 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/1"] "2-2e7-128.101-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.101" [] [[0.5, 0.0]] 1 true true false false -0.12401119872625734 0 2.37054950194748650624241150838 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/101/10"] "2-2e7-128.109-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.19535261716066935 0 0.28998749949822167323518758326 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/4"] "2-2e7-128.109-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.3753333185668557 0 0.940303809647705085763462022330 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/11"] "2-2e7-128.109-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false 0.21092002888136077 0 2.63537221696896834216500824767 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/9"] "2-2e7-128.109-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false 0.3140111651650164 0 2.72757066243122629294792892544 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/8"] "2-2e7-128.109-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false 0.42281830982233404 0 2.73116661081429475937941782681 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/3"] "2-2e7-128.109-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false 0.12401119872625734 0 2.73245226033875988404517243887 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/10"] "2-2e7-128.109-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false 0.37416745284323655 0 3.54951388846334741126157535561 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/12"] "2-2e7-128.109-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.3224080031866884 0 1.42406646491903993672052035116 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/7"] "2-2e7-128.109-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.030422108194951548 0 1.54360325812182180933004949612 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/6"] "2-2e7-128.109-c1-0-4" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false 0.06804628103861464 0 1.71691573623690028525774912696 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/2"] "2-2e7-128.109-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.09375472367893647 0 1.97701274197071231462140371422 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/5"] "2-2e7-128.109-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.02048013331729377 0 2.26574981010771969101681335190 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/13"] "2-2e7-128.109-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.03486794831387463 0 2.36740389418477339427650542073 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/1"] "2-2e7-128.109-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false -0.11538900917735488 0 2.40186077640909097961694043157 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/14"] "2-2e7-128.109-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.109" [] [[0.5, 0.0]] 1 true true false false 0.0412209251198049 0 2.57097917400585825550211148999 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/109/15"] "2-2e7-128.117-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false -0.26313392979765127 0 0.28553268985178812114536558722 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/7"] "2-2e7-128.117-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.48626652496592565 0 0.54774668922782752108278553158 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/8"] "2-2e7-128.117-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false -0.13079523181491073 0 2.63375349267133395125855514353 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/11"] "2-2e7-128.117-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.2114491700107706 0 2.80684857032284140622750117440 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/14"] "2-2e7-128.117-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.1763450107588065 0 2.88143550092494598179745555270 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/9"] "2-2e7-128.117-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.10515732588696276 0 2.89439389428426074727411602204 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/13"] "2-2e7-128.117-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.23375795393812945 0 3.09859031458608895414522313290 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/10"] "2-2e7-128.117-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false -0.17094029450117917 0 0.952842609167884401467733588664 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/3"] "2-2e7-128.117-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false -0.349302087866699 0 1.06325065036128257016635362883 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/12"] "2-2e7-128.117-c1-0-4" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.13129099741936487 0 1.48028221957522895590773804882 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/1"] "2-2e7-128.117-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false -0.17765821448726354 0 1.92526531548062965978639380343 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/6"] "2-2e7-128.117-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.06830439514953592 0 2.12981158143748007574484360709 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/5"] "2-2e7-128.117-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.03962687318504885 0 2.26128478875938596832875328612 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/2"] "2-2e7-128.117-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false -0.0651630286758814 0 2.31774303233298479256069002590 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/15"] "2-2e7-128.117-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.117" [] [[0.5, 0.0]] 1 true true false false 0.27510703582904056 0 2.43775487010288931267888304949 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/117/4"] "2-2e7-128.125-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.08165610948955788 0 1.56112960814841965819628299788 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/5"] "2-2e7-128.125-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.1352221657440676 0 1.57383338390403861899708139134 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/1"] "2-2e7-128.125-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false 0.32947408148919294 0 3.10042409512481927815293761361 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/7"] "2-2e7-128.125-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false 0.49570462305141244 0 3.10643438025720522263650863824 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/2"] "2-2e7-128.125-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false 0.32106901066649784 0 3.42917208586650995550782705759 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/12"] "2-2e7-128.125-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false 0.24050524176959395 0 3.51010807117576159071092088050 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/15"] "2-2e7-128.125-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.496901532439278 0 3.84044786798966247675863329267 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/8"] "2-2e7-128.125-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.29616242647446117 0 1.71461816426515159611921988875 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/10"] "2-2e7-128.125-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.26577630071599384 0 1.79265803953979351910642536650 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/4"] "2-2e7-128.125-c1-0-4" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.04592128791579327 0 1.90266813666587267047146880341 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/9"] "2-2e7-128.125-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false 0.12918709118853122 0 1.97121651659181749164876106540 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/6"] "2-2e7-128.125-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false 0.17758581514831298 0 2.00166952076268306540551776535 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/3"] "2-2e7-128.125-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.09838234905230071 0 2.19289792672276833638239055785 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/13"] "2-2e7-128.125-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false -0.010612011264653807 0 2.68389640302813723021381404344 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/14"] "2-2e7-128.125-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.125" [] [[0.5, 0.0]] 1 true true false false 0.16679581978256497 0 2.72502354188794415347963209393 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/125/11"] "2-2e7-128.13-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.13" [] [[0.5, 0.0]] 1 true true false false -0.11340388120258477 0 1.01737682594702185660343890112 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/13/4"] "2-2e7-128.13-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.13" [] [[0.5, 0.0]] 1 true true false false -0.38945379771882 0 1.22387903811234717769261473325 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/13/7"] "2-2e7-128.13-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.13" [] [[0.5, 0.0]] 1 true true false false 0.006532871638513746 0 2.54984973247980450448237001947 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/13/13"] "2-2e7-128.13-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.13" [] [[0.5, 0.0]] 1 true true false false 0.4139466645755606 0 2.90350509236817510882706937479 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/13/2"] "2-2e7-128.13-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.13" [] [[0.5, 0.0]] 1 true true false false 0.1631073198166365 0 3.08523782158913302076874791224 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/13/9"] "2-2e7-128.13-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.13" [] [[0.5, 0.0]] 1 true true false false 0.3294772821375607 0 3.37547046037886355960071225261 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/13/10"] "2-2e7-128.13-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.13" [] [[0.5, 0.0]] 1 true true false false 0.24542394207147736 0 3.70303605351466838007510823455 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/13/14"] "2-2e7-128.13-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.13" 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[] [[0.5, 0.0]] 1 true true false false -0.3294772821375607 0 1.40686105363148896559352601030 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/69/10"] "2-2e7-128.69-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.69" [] [[0.5, 0.0]] 1 true true false false -0.17759954192863645 0 1.44358193427401358586741102220 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/69/5"] "2-2e7-128.69-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.69" [] [[0.5, 0.0]] 1 true true false false -0.24542394207147736 0 1.83858249099139201992771210461 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/69/14"] "2-2e7-128.69-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.69" [] [[0.5, 0.0]] 1 true true false false 0.11340388120258477 0 2.41122582848701872174028972700 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/69/4"] "2-2e7-128.69-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.69" [] [[0.5, 0.0]] 1 true true false false 0.020797707349244902 0 2.45568230392852285759137422044 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/69/15"] "2-2e7-128.69-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.69" [] [[0.5, 0.0]] 1 true true false false 0.2911072974274818 0 2.65734915189273694774482184923 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/69/3"] "2-2e7-128.77-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.3429343482530965 0 0.31882850789141683406295532923 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/3"] "2-2e7-128.77-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.48844711531084156 0 0.58969194411320244948011975109 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/8"] "2-2e7-128.77-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false 0.23621320271035498 0 2.72401719758557860208387187878 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/10"] "2-2e7-128.77-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.04122076947320444 0 2.72791625158947158688460587345 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/14"] "2-2e7-128.77-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false 0.11865518907731175 0 2.86378999261687741543908074781 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/13"] "2-2e7-128.77-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false 0.22900854830806208 0 3.46278347765940815335523938179 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/9"] "2-2e7-128.77-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.4229998001833183 0 4.09558463715149325293245922798 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/12"] "2-2e7-128.77-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.2806146121615817 0 0.866140296000294150071270814141 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/2"] "2-2e7-128.77-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false 0.09427617015781549 0 1.37920037137844632028709295664 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/5"] "2-2e7-128.77-c1-0-4" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.10806901287380026 0 1.96859285681459398041203482491 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/15"] "2-2e7-128.77-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.17083457791082057 0 2.12821864728828948114310199424 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/11"] "2-2e7-128.77-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false 0.20367922212443948 0 2.13893040580769448760097278067 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/4"] "2-2e7-128.77-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false 0.050443726319983846 0 2.14013934896658316799742970247 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/1"] "2-2e7-128.77-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false 0.1233933321439329 0 2.14814531351906436151292046336 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/6"] "2-2e7-128.77-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.77" [] [[0.5, 0.0]] 1 true true false false -0.08336165467523726 0 2.31705046954492786217967753569 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/77/7"] "2-2e7-128.85-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false -0.32947408148919294 0 0.21858111278806445908513386427 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/7"] "2-2e7-128.85-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false -0.49570462305141244 0 0.23212895176522599729842231963 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/2"] "2-2e7-128.85-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.08165610948955788 0 2.64852337642421032358265353293 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/5"] "2-2e7-128.85-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.09838234905230071 0 2.68937134504149217102459262557 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/13"] "2-2e7-128.85-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.26577630071599384 0 2.70987965890438920213039077847 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/4"] "2-2e7-128.85-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.010612011264653807 0 3.03072831371374738842670836146 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/14"] "2-2e7-128.85-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.29616242647446117 0 3.51848139762504523511518583613 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/10"] "2-2e7-128.85-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.496901532439278 0 0.59958836192224943836794559833 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/8"] "2-2e7-128.85-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false -0.17758581514831298 0 1.21470281953699166599575346525 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/3"] "2-2e7-128.85-c1-0-4" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false -0.24050524176959395 0 1.41151256741134945665970551071 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/15"] "2-2e7-128.85-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false -0.32106901066649784 0 1.47643746838582625325611347995 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/12"] "2-2e7-128.85-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.04592128791579327 0 1.66044355732207204913342680172 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/9"] "2-2e7-128.85-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false -0.12918709118853122 0 1.90011501166047986097876516586 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/6"] "2-2e7-128.85-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false -0.16679581978256497 0 1.94665497240407132425496047980 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/11"] "2-2e7-128.85-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.85" [] [[0.5, 0.0]] 1 true true false false 0.1352221657440676 0 2.35169178852170289386221233784 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/85/1"] "2-2e7-128.93-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.13129099741936487 0 0.76067201722654421154937560173 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/1"] "2-2e7-128.93-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.06830439514953592 0 1.03130998518253711823781677430 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/5"] "2-2e7-128.93-c1-0-10" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false 0.13079523181491073 0 2.67057194286562910991897782974 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/11"] "2-2e7-128.93-c1-0-11" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false 0.0651630286758814 0 2.74807047629660629955399762809 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/15"] "2-2e7-128.93-c1-0-12" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false 0.17094029450117917 0 2.77462595998485998919839790002 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/3"] "2-2e7-128.93-c1-0-13" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false 0.349302087866699 0 3.28271939916377329261488630985 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/12"] "2-2e7-128.93-c1-0-14" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.48626652496592565 0 3.61515596763959349715060226890 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/8"] "2-2e7-128.93-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.27510703582904056 0 1.22200290279759522243078304830 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/4"] "2-2e7-128.93-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.2114491700107706 0 1.38516080503868109346726010401 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/14"] "2-2e7-128.93-c1-0-4" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.03962687318504885 0 1.81924403885291380966268015669 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/2"] "2-2e7-128.93-c1-0-5" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.23375795393812945 0 1.92605121697829696444695927255 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/10"] "2-2e7-128.93-c1-0-6" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.1763450107588065 0 2.31338718530782566169368189669 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/9"] "2-2e7-128.93-c1-0-7" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false -0.10515732588696276 0 2.38807060304961299325073386526 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/13"] "2-2e7-128.93-c1-0-8" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false 0.26313392979765127 0 2.51689614740104881647275588144 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/7"] "2-2e7-128.93-c1-0-9" 1.0109822678328189 1.0220851458723894 2 128 "128.93" [] [[0.5, 0.0]] 1 true true false false 0.17765821448726354 0 2.55948027610174796525883142001 ["ModularForm/GL2/Q/holomorphic/128/2/k/a/93/6"] "2-2e7-16.13-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "16.13" [] [[0.5, 0.0]] 1 true true false false -0.06250000000000001 0 2.03247494786444624826600944225 ["ModularForm/GL2/Q/holomorphic/128/2/e/b/33/1"] "2-2e7-16.13-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "16.13" [] [[0.5, 0.0]] 1 true true false false 0.1875 0 2.56170305732102056983156404427 ["ModularForm/GL2/Q/holomorphic/128/2/e/a/33/1"] "2-2e7-16.5-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "16.5" [] [[0.5, 0.0]] 1 true true false false -0.1875 0 1.29307975640473051161799074505 ["ModularForm/GL2/Q/holomorphic/128/2/e/a/97/1"] "2-2e7-16.5-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "16.5" [] [[0.5, 0.0]] 1 true true false false 0.06250000000000001 0 2.40795688562533763666136812839 ["ModularForm/GL2/Q/holomorphic/128/2/e/b/97/1"] "2-2e7-32.13-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "32.13" [] [[0.5, 0.0]] 1 true true false false -0.34375000000000006 0 0.802755396945065606216234866833 ["ModularForm/GL2/Q/holomorphic/128/2/g/a/17/1"] "2-2e7-32.13-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "32.13" [] [[0.5, 0.0]] 1 true true false false 0.0038377421589559243 0 2.04034369755235839957816958855 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/17/1"] "2-2e7-32.13-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "32.13" [] [[0.5, 0.0]] 1 true true false false 0.1836622578410441 0 3.04804304390972786915897681540 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/17/2"] "2-2e7-32.21-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "32.21" [] [[0.5, 0.0]] 1 true true false false -0.21291785495867066 0 1.10728697400188391353441493973 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/49/1"] "2-2e7-32.21-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "32.21" [] [[0.5, 0.0]] 1 true true false false 0.025417854958670657 0 2.41943147861075777724173835435 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/49/2"] "2-2e7-32.21-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "32.21" [] [[0.5, 0.0]] 1 true true false false 0.09375 0 2.62267687397708442152110336678 ["ModularForm/GL2/Q/holomorphic/128/2/g/a/49/1"] "2-2e7-32.29-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "32.29" [] [[0.5, 0.0]] 1 true true false false -0.09375 0 1.78087375666559412677570346116 ["ModularForm/GL2/Q/holomorphic/128/2/g/a/81/1"] "2-2e7-32.29-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "32.29" [] [[0.5, 0.0]] 1 true true false false -0.025417854958670657 0 2.44255553727107700784220926850 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/81/2"] "2-2e7-32.29-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "32.29" [] [[0.5, 0.0]] 1 true true false false 0.21291785495867066 0 2.57619057750818061456918049157 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/81/1"] "2-2e7-32.5-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "32.5" [] [[0.5, 0.0]] 1 true true false false -0.1836622578410441 0 1.75381149472773307070969774970 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/113/2"] "2-2e7-32.5-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "32.5" [] [[0.5, 0.0]] 1 true true false false -0.0038377421589559243 0 1.96427462206719274107009010982 ["ModularForm/GL2/Q/holomorphic/128/2/g/b/113/1"] "2-2e7-32.5-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "32.5" [] [[0.5, 0.0]] 1 true true false false 0.34375000000000006 0 3.36055108308096228739991492130 ["ModularForm/GL2/Q/holomorphic/128/2/g/a/113/1"] "2-2e7-8.5-c1-0-0" 1.0109822678328189 1.0220851458723894 2 128 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.25 0 1.38179389167272920808910458982 ["ModularForm/GL2/Q/holomorphic/128/2/b/a/65/2"] "2-2e7-8.5-c1-0-1" 1.0109822678328189 1.0220851458723894 2 128 "8.5" [] [[0.5, 0.0]] 1 true true false false -0.125 0 1.52019857626457482641329780176 ["ModularForm/GL2/Q/holomorphic/128/2/b/b/65/2"] "2-2e7-8.5-c1-0-2" 1.0109822678328189 1.0220851458723894 2 128 "8.5" [] [[0.5, 0.0]] 1 true true false false 0.125 0 2.57482920414217144195647823211 ["ModularForm/GL2/Q/holomorphic/128/2/b/b/65/1"] "2-2e7-8.5-c1-0-3" 1.0109822678328189 1.0220851458723894 2 128 "8.5" [] [[0.5, 0.0]] 1 true true false false 0.25 0 3.08634036933586957896715177110 ["ModularForm/GL2/Q/holomorphic/128/2/b/a/65/1"] # 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 $1$. #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).