# lfunc_search downloaded from the LMFDB on 15 July 2026. # Search link: https://www.lmfdb.org/L/2/3096/129.128/c1-0 # Query "{'degree': 2, 'conductor': 3096, 'spectral_label': 'c1-0'}" returned 185 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. "2-3096-1.1-c1-0-0" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.25603509802084131310488446021 ["ModularForm/GL2/Q/holomorphic/3096/2/a/v/1/1"] "2-3096-1.1-c1-0-1" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.28456311902447236698528618069 ["ModularForm/GL2/Q/holomorphic/3096/2/a/t/1/2"] "2-3096-1.1-c1-0-10" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.811176660241764713201996000524 ["ModularForm/GL2/Q/holomorphic/3096/2/a/t/1/1"] "2-3096-1.1-c1-0-11" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.841512604448996731513887279789 ["EllipticCurve/Q/3096/a", "ModularForm/GL2/Q/holomorphic/3096/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/a"] "2-3096-1.1-c1-0-12" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.846517227237974718380896109130 ["ModularForm/GL2/Q/holomorphic/3096/2/a/r/1/2"] "2-3096-1.1-c1-0-13" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.876744620883957602821525318359 ["ModularForm/GL2/Q/holomorphic/3096/2/a/v/1/2"] "2-3096-1.1-c1-0-14" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.903025082511793020836717695820 ["EllipticCurve/Q/3096/c", "ModularForm/GL2/Q/holomorphic/3096/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/c"] "2-3096-1.1-c1-0-15" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.931994669818592012873005113940 ["ModularForm/GL2/Q/holomorphic/3096/2/a/s/1/2"] "2-3096-1.1-c1-0-16" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.967216372268755633794991928306 ["ModularForm/GL2/Q/holomorphic/3096/2/a/q/1/3"] "2-3096-1.1-c1-0-17" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.979627824143308013603081205719 ["ModularForm/GL2/Q/holomorphic/3096/2/a/i/1/2"] "2-3096-1.1-c1-0-18" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.986164057574362934972196520372 ["ModularForm/GL2/Q/holomorphic/3096/2/a/v/1/4"] "2-3096-1.1-c1-0-19" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.989068742462070663431990036877 ["ModularForm/GL2/Q/holomorphic/3096/2/a/r/1/3"] "2-3096-1.1-c1-0-2" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.42806483268838095657486950855 ["ModularForm/GL2/Q/holomorphic/3096/2/a/s/1/1"] "2-3096-1.1-c1-0-20" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.03698288676849615429422409406 ["ModularForm/GL2/Q/holomorphic/3096/2/a/m/1/2"] "2-3096-1.1-c1-0-21" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.05517951300682127311566861960 ["ModularForm/GL2/Q/holomorphic/3096/2/a/v/1/6"] "2-3096-1.1-c1-0-22" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11290499826596340875002009979 ["ModularForm/GL2/Q/holomorphic/3096/2/a/l/1/2"] "2-3096-1.1-c1-0-23" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.13139707483970270013360943176 ["ModularForm/GL2/Q/holomorphic/3096/2/a/t/1/4"] "2-3096-1.1-c1-0-24" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.19960700624628580791342369988 ["ModularForm/GL2/Q/holomorphic/3096/2/a/u/1/1"] "2-3096-1.1-c1-0-25" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.21143804372333352703419626567 ["ModularForm/GL2/Q/holomorphic/3096/2/a/q/1/2"] "2-3096-1.1-c1-0-26" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.22678857003881424840587152812 ["ModularForm/GL2/Q/holomorphic/3096/2/a/n/1/1"] "2-3096-1.1-c1-0-27" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.24204565960065278427357421554 ["EllipticCurve/Q/3096/g", "ModularForm/GL2/Q/holomorphic/3096/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/g"] "2-3096-1.1-c1-0-28" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.25207923777746649346719521093 ["ModularForm/GL2/Q/holomorphic/3096/2/a/v/1/5"] "2-3096-1.1-c1-0-29" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.28786972539082923160859583130 ["ModularForm/GL2/Q/holomorphic/3096/2/a/o/1/1"] "2-3096-1.1-c1-0-3" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.53263568651362052684770248510 ["ModularForm/GL2/Q/holomorphic/3096/2/a/m/1/1"] "2-3096-1.1-c1-0-30" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.29302321454148338892057587490 ["ModularForm/GL2/Q/holomorphic/3096/2/a/p/1/1"] "2-3096-1.1-c1-0-31" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.30963542335610990783092633770 ["ModularForm/GL2/Q/holomorphic/3096/2/a/t/1/5"] "2-3096-1.1-c1-0-32" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.33951079569544540632246002438 ["ModularForm/GL2/Q/holomorphic/3096/2/a/s/1/3"] "2-3096-1.1-c1-0-33" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.35011573591576701379135886180 ["ModularForm/GL2/Q/holomorphic/3096/2/a/u/1/2"] "2-3096-1.1-c1-0-34" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.37488185040860504773382864563 ["ModularForm/GL2/Q/holomorphic/3096/2/a/n/1/2"] "2-3096-1.1-c1-0-35" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.39013289600463572921512800259 ["ModularForm/GL2/Q/holomorphic/3096/2/a/u/1/4"] "2-3096-1.1-c1-0-36" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.42323441342017630499172507901 ["ModularForm/GL2/Q/holomorphic/3096/2/a/o/1/2"] "2-3096-1.1-c1-0-37" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.43046628272075442915585224740 ["ModularForm/GL2/Q/holomorphic/3096/2/a/j/1/1"] "2-3096-1.1-c1-0-38" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.46621268900372359483904829364 ["ModularForm/GL2/Q/holomorphic/3096/2/a/j/1/2"] "2-3096-1.1-c1-0-39" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.50480930537648516900015840183 ["ModularForm/GL2/Q/holomorphic/3096/2/a/p/1/2"] "2-3096-1.1-c1-0-4" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.55105365902456005462272029894 ["ModularForm/GL2/Q/holomorphic/3096/2/a/r/1/1"] "2-3096-1.1-c1-0-40" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.54068274205403971934956822625 ["ModularForm/GL2/Q/holomorphic/3096/2/a/k/1/1"] "2-3096-1.1-c1-0-41" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.54213231540505679095547676326 ["EllipticCurve/Q/3096/d", "ModularForm/GL2/Q/holomorphic/3096/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/d"] "2-3096-1.1-c1-0-42" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.55060584877535680363479116348 ["EllipticCurve/Q/3096/b", "ModularForm/GL2/Q/holomorphic/3096/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/b"] "2-3096-1.1-c1-0-43" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.61416356536828273616795186724 ["ModularForm/GL2/Q/holomorphic/3096/2/a/u/1/3"] "2-3096-1.1-c1-0-44" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.63770359422184508311135901286 ["EllipticCurve/Q/3096/e", "ModularForm/GL2/Q/holomorphic/3096/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/e"] "2-3096-1.1-c1-0-45" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.63940361235119110070795121935 ["ModularForm/GL2/Q/holomorphic/3096/2/a/u/1/6"] "2-3096-1.1-c1-0-46" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.65249073355749917265913176364 ["ModularForm/GL2/Q/holomorphic/3096/2/a/u/1/5"] "2-3096-1.1-c1-0-47" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.65261880495802064667134938617 ["EllipticCurve/Q/3096/h", "ModularForm/GL2/Q/holomorphic/3096/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/h"] "2-3096-1.1-c1-0-48" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.68519082747847804410458688381 ["ModularForm/GL2/Q/holomorphic/3096/2/a/p/1/3"] "2-3096-1.1-c1-0-49" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.70290744015271455898842739385 ["ModularForm/GL2/Q/holomorphic/3096/2/a/k/1/2"] "2-3096-1.1-c1-0-5" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.59009606634214281358086081863 ["ModularForm/GL2/Q/holomorphic/3096/2/a/v/1/3"] "2-3096-1.1-c1-0-50" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.74558122029121200555752621699 ["EllipticCurve/Q/3096/f", "ModularForm/GL2/Q/holomorphic/3096/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/3096/2/a/f"] "2-3096-1.1-c1-0-51" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.79077292615424791948153253509 ["ModularForm/GL2/Q/holomorphic/3096/2/a/n/1/3"] "2-3096-1.1-c1-0-52" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.83083072885499275828381899071 ["ModularForm/GL2/Q/holomorphic/3096/2/a/o/1/3"] "2-3096-1.1-c1-0-6" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.59777697168721306047191259980 ["ModularForm/GL2/Q/holomorphic/3096/2/a/t/1/3"] "2-3096-1.1-c1-0-7" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.61371590860522378940128860501 ["ModularForm/GL2/Q/holomorphic/3096/2/a/q/1/1"] "2-3096-1.1-c1-0-8" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.63975906579214953977401778674 ["ModularForm/GL2/Q/holomorphic/3096/2/a/i/1/1"] "2-3096-1.1-c1-0-9" 4.972090552854847 24.721684465788417 2 3096 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.73445343808585114560585955570 ["ModularForm/GL2/Q/holomorphic/3096/2/a/l/1/1"] "2-3096-129.128-c1-0-0" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.3188771967780346 0 0.07473494786276159258711836486 ["ModularForm/GL2/Q/holomorphic/3096/2/l/e/2321/3"] "2-3096-129.128-c1-0-1" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.384234647560612 0 0.13029252288369839322093714650 ["ModularForm/GL2/Q/holomorphic/3096/2/l/f/2321/9"] "2-3096-129.128-c1-0-10" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.2934045160811978 0 0.44572767840186274944935090753 ["ModularForm/GL2/Q/holomorphic/3096/2/l/e/2321/12"] "2-3096-129.128-c1-0-11" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.03850680920211301 0 0.51313141202647799545772518417 ["ModularForm/GL2/Q/holomorphic/3096/2/l/f/2321/3"] "2-3096-129.128-c1-0-12" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.23596902075550577 0 0.51382182241077380234304926923 ["ModularForm/GL2/Q/holomorphic/3096/2/l/c/2321/1"] "2-3096-129.128-c1-0-13" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false 0.03850680920211301 0 0.60638049997524508167933512372 ["ModularForm/GL2/Q/holomorphic/3096/2/l/f/2321/4"] "2-3096-129.128-c1-0-14" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.08802479519156195 0 0.61878235391860496505017011560 ["ModularForm/GL2/Q/holomorphic/3096/2/l/e/2321/7"] "2-3096-129.128-c1-0-15" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.040055744740202126 0 0.64096002281387407379303769302 ["ModularForm/GL2/Q/holomorphic/3096/2/l/a/2321/2"] "2-3096-129.128-c1-0-16" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.23596902075550577 0 0.65479805162101563477870990297 ["ModularForm/GL2/Q/holomorphic/3096/2/l/d/2321/2"] "2-3096-129.128-c1-0-17" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.15740646681319065 0 0.67333394983474209521484861451 ["ModularForm/GL2/Q/holomorphic/3096/2/l/e/2321/16"] "2-3096-129.128-c1-0-18" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.12296392076273095 0 0.71808872327446364160391965043 ["ModularForm/GL2/Q/holomorphic/3096/2/l/f/2321/15"] "2-3096-129.128-c1-0-19" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false -0.040055744740202126 0 0.73565923062395849547835092222 ["ModularForm/GL2/Q/holomorphic/3096/2/l/b/2321/2"] "2-3096-129.128-c1-0-2" 4.972090552854847 24.721684465788417 2 3096 "129.128" [] [[0.5, 0.0]] 1 true true false false 0.395150228816615 0 0.16351776709559787326625892411 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false false -0.16544256757473028 0 0.41629723231358686879027000365 ["ModularForm/GL2/Q/holomorphic/3096/2/bm/a/1241/17"] # 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).