# lfunc_search downloaded from the LMFDB on 30 April 2026. # Search link: https://www.lmfdb.org/L/rational/2/15 # Query "{'degree': 2, 'conductor': 15, 'rational': True}" returned 62 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, $\epsilon$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-15-1.1-c1-0-0" 0.34608612083110274 0.11977560303192064 2 15 "1.1" [] [[0.5, 0.0]] 1 true 1 0 5.23920392624592057055772361749 ["EllipticCurve/Q/15/a", "ModularForm/GL2/Q/holomorphic/15/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/2/a/a"] "2-15-15.14-c2-0-0" 0.6393124404705772 0.4087203965404454 2 15 "15.14" [] [[1.0, 0.0]] 2 true 1 0 4.08039726064136068182425159874 ["ModularForm/GL2/Q/holomorphic/15/3/d/a/14/1", "ModularForm/GL2/Q/holomorphic/15/3/d/a"] "2-15-15.14-c2-0-1" 0.6393124404705772 0.4087203965404454 2 15 "15.14" [] [[1.0, 0.0]] 2 true 1 0 4.84192581422996258804553371125 ["ModularForm/GL2/Q/holomorphic/15/3/d/b/14/1", "ModularForm/GL2/Q/holomorphic/15/3/d/b"] "2-15-1.1-c3-0-0" 0.9407596133370679 0.8850286500861096 2 15 "1.1" [] [[1.5, 0.0]] 3 true 1 0 3.78175129911665234242031093429 ["ModularForm/GL2/Q/holomorphic/15/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/4/a/a"] "2-15-1.1-c3-0-1" 0.9407596133370679 0.8850286500861096 2 15 "1.1" [] [[1.5, 0.0]] 3 true 1 0 4.25806719846960648281623011472 ["ModularForm/GL2/Q/holomorphic/15/4/a/b/1/1", "ModularForm/GL2/Q/holomorphic/15/4/a/b"] "2-15-15.14-c4-0-1" 1.2452106031742844 1.5505494462576654 2 15 "15.14" [] [[2.0, 0.0]] 4 true 1 0 2.06476355225246860341332385126 ["ModularForm/GL2/Q/holomorphic/15/5/d/a/14/1", "ModularForm/GL2/Q/holomorphic/15/5/d/a"] "2-15-15.14-c4-0-4" 1.2452106031742844 1.5505494462576654 2 15 "15.14" [] [[2.0, 0.0]] 4 true 1 0 3.96476322029534811803044665594 ["ModularForm/GL2/Q/holomorphic/15/5/d/b/14/1", "ModularForm/GL2/Q/holomorphic/15/5/d/b"] "2-15-1.1-c5-0-1" 1.5510503851244517 2.40575729719471 2 15 "1.1" [] [[2.5, 0.0]] 5 true 1 0 3.28780360799114787730035245347 ["ModularForm/GL2/Q/holomorphic/15/6/a/b/1/1", "ModularForm/GL2/Q/holomorphic/15/6/a/b"] "2-15-1.1-c5-0-3" 1.5510503851244517 2.40575729719471 2 15 "1.1" [] [[2.5, 0.0]] 5 true -1 1 4.35806742067215223865182507911 ["ModularForm/GL2/Q/holomorphic/15/6/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/6/a/a"] "2-15-15.14-c6-0-2" 1.8576359315801805 3.4508112542977654 2 15 "15.14" [] [[3.0, 0.0]] 6 true 1 0 1.17534639145905483763567441638 ["ModularForm/GL2/Q/holomorphic/15/7/d/a/14/1", "ModularForm/GL2/Q/holomorphic/15/7/d/a"] "2-15-15.14-c6-0-7" 1.8576359315801805 3.4508112542977654 2 15 "15.14" [] [[3.0, 0.0]] 6 true 1 0 3.13495099498879613973177589185 ["ModularForm/GL2/Q/holomorphic/15/7/d/b/14/1", "ModularForm/GL2/Q/holomorphic/15/7/d/b"] "2-15-1.1-c7-0-0" 2.1646651894142495 4.685775382261828 2 15 "1.1" [] [[3.5, 0.0]] 7 true 1 0 1.08668704122281820100157322467 ["ModularForm/GL2/Q/holomorphic/15/8/a/b/1/1", "ModularForm/GL2/Q/holomorphic/15/8/a/b"] "2-15-1.1-c7-0-2" 2.1646651894142495 4.685775382261828 2 15 "1.1" [] [[3.5, 0.0]] 7 true -1 1 2.47871564718276244394021493538 ["ModularForm/GL2/Q/holomorphic/15/8/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/8/a/a"] "2-15-15.14-c8-0-3" 2.4719787925702175 6.11067915091691 2 15 "15.14" [] [[4.0, 0.0]] 8 true 1 0 0.73359940383405495170141714605 ["ModularForm/GL2/Q/holomorphic/15/9/d/a/14/1", "ModularForm/GL2/Q/holomorphic/15/9/d/a"] "2-15-15.14-c8-0-9" 2.4719787925702175 6.11067915091691 2 15 "15.14" [] [[4.0, 0.0]] 8 true 1 0 2.39464132592368190989060430507 ["ModularForm/GL2/Q/holomorphic/15/9/d/b/14/1", "ModularForm/GL2/Q/holomorphic/15/9/d/b"] "2-15-1.1-c9-0-4" 2.77948512182708 7.725537542458099 2 15 "1.1" [] [[4.5, 0.0]] 9 true -1 1 2.89046851552116422920781524476 ["ModularForm/GL2/Q/holomorphic/15/10/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/10/a/a"] "2-15-1.1-c9-0-5" 2.77948512182708 7.725537542458099 2 15 "1.1" [] [[4.5, 0.0]] 9 true -1 1 3.44606926321203345768029167895 ["ModularForm/GL2/Q/holomorphic/15/10/a/b/1/1", "ModularForm/GL2/Q/holomorphic/15/10/a/b"] "2-15-15.14-c10-0-13" 3.08712791929746 9.530358790105867 2 15 "15.14" [] [[5.0, 0.0]] 10 true 1 0 2.05477476912722704308373511457 ["ModularForm/GL2/Q/holomorphic/15/11/d/b/14/1", "ModularForm/GL2/Q/holomorphic/15/11/d/b"] "2-15-15.14-c10-0-7" 3.08712791929746 9.530358790105867 2 15 "15.14" [] [[5.0, 0.0]] 10 true 1 0 1.04101687092716463084980436862 ["ModularForm/GL2/Q/holomorphic/15/11/d/a/14/1", "ModularForm/GL2/Q/holomorphic/15/11/d/a"] "2-15-1.1-c11-0-4" 3.3948707940647513 11.525147708393837 2 15 "1.1" [] [[5.5, 0.0]] 11 true -1 1 1.74079702699032931824273264204 ["ModularForm/GL2/Q/holomorphic/15/12/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/12/a/a"] "2-15-15.14-c12-0-13" 3.7026891928899848 13.709907259144286 2 15 "15.14" [] [[6.0, 0.0]] 12 true 1 0 1.35949969895292649148116130502 ["ModularForm/GL2/Q/holomorphic/15/13/d/a/14/1", "ModularForm/GL2/Q/holomorphic/15/13/d/a"] "2-15-15.14-c12-0-6" 3.7026891928899848 13.709907259144286 2 15 "15.14" [] [[6.0, 0.0]] 12 true 1 0 0.62640192984036775781604074273 ["ModularForm/GL2/Q/holomorphic/15/13/d/b/14/1", "ModularForm/GL2/Q/holomorphic/15/13/d/b"] "2-15-1.1-c13-0-6" 4.010565962901859 16.084639342786915 2 15 "1.1" [] [[6.5, 0.0]] 13 true -1 1 1.94220697433054868390112595442 ["ModularForm/GL2/Q/holomorphic/15/14/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/14/a/a"] "2-15-15.14-c14-0-21" 4.318488766083097 18.649345222785907 2 15 "15.14" [] [[7.0, 0.0]] 14 true 1 0 2.13572071330093999255751324822 ["ModularForm/GL2/Q/holomorphic/15/15/d/b/14/1", "ModularForm/GL2/Q/holomorphic/15/15/d/b"] "2-15-15.14-c14-0-9" 4.318488766083097 18.649345222785907 2 15 "15.14" [] [[7.0, 0.0]] 14 true 1 0 0.76345598695325543615704315232 ["ModularForm/GL2/Q/holomorphic/15/15/d/a/14/1", "ModularForm/GL2/Q/holomorphic/15/15/d/a"] "2-15-15.14-c16-0-12" 4.934438324520236 24.34868157849407 2 15 "15.14" [] [[8.0, 0.0]] 16 true 1 0 0.69263199375875268171424722668 ["ModularForm/GL2/Q/holomorphic/15/17/d/b", "ModularForm/GL2/Q/holomorphic/15/17/d/b/14/1"] "2-15-15.14-c16-0-17" 4.934438324520236 24.34868157849407 2 15 "15.14" [] [[8.0, 0.0]] 16 true 1 0 1.09767126616123555709837102322 ["ModularForm/GL2/Q/holomorphic/15/17/d/a", "ModularForm/GL2/Q/holomorphic/15/17/d/a/14/1"] "2-15-15.14-c18-0-14" 5.550488325946154 30.807920656464542 2 15 "15.14" [] [[9.0, 0.0]] 18 true 1 0 0.74577549594299030929790711482 ["ModularForm/GL2/Q/holomorphic/15/19/d/a", "ModularForm/GL2/Q/holomorphic/15/19/d/a/14/1"] "2-15-15.14-c18-0-21" 5.550488325946154 30.807920656464542 2 15 "15.14" [] [[9.0, 0.0]] 18 true 1 0 1.22451104414802360124942768687 ["ModularForm/GL2/Q/holomorphic/15/19/d/b", "ModularForm/GL2/Q/holomorphic/15/19/d/b/14/1"] "2-15-15.14-c20-0-3" 6.1666088545812014 38.027064765399274 2 15 "15.14" [] [[10.0, 0.0]] 20 true 1 0 0.24257493012393896255639220120 ["ModularForm/GL2/Q/holomorphic/15/21/d/a", "ModularForm/GL2/Q/holomorphic/15/21/d/a/14/1"] "2-15-15.14-c20-0-30" 6.1666088545812014 38.027064765399274 2 15 "15.14" [] [[10.0, 0.0]] 20 true 1 0 1.78537331329145048014099824751 ["ModularForm/GL2/Q/holomorphic/15/21/d/b", "ModularForm/GL2/Q/holomorphic/15/21/d/b/14/1"] "2-15-1.1-c21-0-6" 6.474689308612369 41.92160164305932 2 15 "1.1" [] [[10.5, 0.0]] 21 true 1 0 0.908762802151516217531955448285 ["ModularForm/GL2/Q/holomorphic/15/22/a/a/1/1", "ModularForm/GL2/Q/holomorphic/15/22/a/a"] "2-15-15.14-c22-0-15" 6.782780788605627 46.00611522627757 2 15 "15.14" [] [[11.0, 0.0]] 22 true 1 0 0.62382273474032363478193318931 ["ModularForm/GL2/Q/holomorphic/15/23/d/a", "ModularForm/GL2/Q/holomorphic/15/23/d/a/14/1"] "2-15-15.14-c22-0-24" 6.782780788605627 46.00611522627757 2 15 "15.14" [] [[11.0, 0.0]] 22 true 1 0 0.915321926676101081265847712922 ["ModularForm/GL2/Q/holomorphic/15/23/d/b", "ModularForm/GL2/Q/holomorphic/15/23/d/b/14/1"] "2-15-15.14-c24-0-29" 7.398991339277705 54.74507283870649 2 15 "15.14" [] [[12.0, 0.0]] 24 true 1 0 0.966448508222906064978973368798 ["ModularForm/GL2/Q/holomorphic/15/25/d/a", "ModularForm/GL2/Q/holomorphic/15/25/d/a/14/1"] "2-15-15.14-c24-0-31" 7.398991339277705 54.74507283870649 2 15 "15.14" [] [[12.0, 0.0]] 24 true 1 0 1.08793500955262502783059938238 ["ModularForm/GL2/Q/holomorphic/15/25/d/b", "ModularForm/GL2/Q/holomorphic/15/25/d/b/14/1"] "2-15-15.14-c26-0-18" 8.015231631678557 64.2439381094605 2 15 "15.14" [] [[13.0, 0.0]] 26 true 1 0 0.60389580030631224682203098118 ["ModularForm/GL2/Q/holomorphic/15/27/d/a", "ModularForm/GL2/Q/holomorphic/15/27/d/a/14/1"] "2-15-15.14-c26-0-22" 8.015231631678557 64.2439381094605 2 15 "15.14" [] [[13.0, 0.0]] 26 true 1 0 0.67821767770616802104097691562 ["ModularForm/GL2/Q/holomorphic/15/27/d/b", "ModularForm/GL2/Q/holomorphic/15/27/d/b/14/1"] "2-15-15.14-c28-0-25" 8.631495314963589 74.50271137223837 2 15 "15.14" [] [[14.0, 0.0]] 28 true 1 0 0.58544886071435058121225778297 ["ModularForm/GL2/Q/holomorphic/15/29/d/a", "ModularForm/GL2/Q/holomorphic/15/29/d/a/14/1"] "2-15-15.14-c28-0-46" 8.631495314963589 74.50271137223837 2 15 "15.14" [] [[14.0, 0.0]] 28 true 1 0 1.72647782362026491826657470406 ["ModularForm/GL2/Q/holomorphic/15/29/d/b", "ModularForm/GL2/Q/holomorphic/15/29/d/b/14/1"] "2-15-15.14-c30-0-21" 9.247777725161072 85.52139285398529 2 15 "15.14" [] [[15.0, 0.0]] 30 true 1 0 0.61373745422302295759923545268 ["ModularForm/GL2/Q/holomorphic/15/31/d/a", "ModularForm/GL2/Q/holomorphic/15/31/d/a/14/1"] "2-15-15.14-c30-0-33" 9.247777725161072 85.52139285398529 2 15 "15.14" [] [[15.0, 0.0]] 30 true 1 0 0.799222705390201462796597287746 ["ModularForm/GL2/Q/holomorphic/15/31/d/b", "ModularForm/GL2/Q/holomorphic/15/31/d/b/14/1"] "2-15-15.14-c32-0-16" 9.864075360284552 97.29998271337283 2 15 "15.14" [] [[16.0, 0.0]] 32 true 1 0 0.38821747222722001284122512430 ["ModularForm/GL2/Q/holomorphic/15/33/d/b", "ModularForm/GL2/Q/holomorphic/15/33/d/b/14/1"] "2-15-15.14-c32-0-40" 9.864075360284552 97.29998271337283 2 15 "15.14" [] [[16.0, 0.0]] 32 true 1 0 0.911868528614877137789061092177 ["ModularForm/GL2/Q/holomorphic/15/33/d/a", "ModularForm/GL2/Q/holomorphic/15/33/d/a/14/1"] "2-15-15.14-c34-0-32" 10.480385539856178 109.83848106402647 2 15 "15.14" [] [[17.0, 0.0]] 34 true 1 0 0.67480363475196286399791994967 ["ModularForm/GL2/Q/holomorphic/15/35/d/a", "ModularForm/GL2/Q/holomorphic/15/35/d/a/14/1"] "2-15-15.14-c34-0-53" 10.480385539856178 109.83848106402647 2 15 "15.14" [] [[17.0, 0.0]] 34 true 1 0 1.23443202725930949385393665293 ["ModularForm/GL2/Q/holomorphic/15/35/d/b", "ModularForm/GL2/Q/holomorphic/15/35/d/b/14/1"] "2-15-15.14-c36-0-10" 11.096706177467055 123.1368879890355 2 15 "15.14" [] [[18.0, 0.0]] 36 true 1 0 0.25455438342883960415486211581 ["ModularForm/GL2/Q/holomorphic/15/37/d/a", "ModularForm/GL2/Q/holomorphic/15/37/d/a/14/1"] "2-15-15.14-c36-0-49" 11.096706177467055 123.1368879890355 2 15 "15.14" [] [[18.0, 0.0]] 36 true 1 0 0.874268615430289154552959459334 ["ModularForm/GL2/Q/holomorphic/15/37/d/b", "ModularForm/GL2/Q/holomorphic/15/37/d/b/14/1"] "2-15-15.14-c38-0-22" 11.713035624904933 137.1952035502921 2 15 "15.14" [] [[19.0, 0.0]] 38 true 1 0 0.41910418450443958319059365027 ["ModularForm/GL2/Q/holomorphic/15/39/d/a", "ModularForm/GL2/Q/holomorphic/15/39/d/a/14/1"] "2-15-15.14-c38-0-60" 11.713035624904933 137.1952035502921 2 15 "15.14" [] [[19.0, 0.0]] 38 true 1 0 1.19901263665092022633301869107 ["ModularForm/GL2/Q/holomorphic/15/39/d/b", "ModularForm/GL2/Q/holomorphic/15/39/d/b/14/1"] "2-15-15.14-c40-0-43" 12.329372562894731 152.01342779466137 2 15 "15.14" [] [[20.0, 0.0]] 40 true 1 0 0.71362073029840494052280575251 ["ModularForm/GL2/Q/holomorphic/15/41/d/a", "ModularForm/GL2/Q/holomorphic/15/41/d/a/14/1"] "2-15-15.14-c40-0-54" 12.329372562894731 152.01342779466137 2 15 "15.14" [] [[20.0, 0.0]] 40 true 1 0 0.890628041192290035410900798132 ["ModularForm/GL2/Q/holomorphic/15/41/d/b", "ModularForm/GL2/Q/holomorphic/15/41/d/b/14/1"] "2-15-15.14-c42-0-27" 12.945715922966716 167.59156075815397 2 15 "15.14" [] [[21.0, 0.0]] 42 true 1 0 0.42245212917345598073130685026 ["ModularForm/GL2/Q/holomorphic/15/43/d/a", "ModularForm/GL2/Q/holomorphic/15/43/d/a/14/1"] "2-15-15.14-c42-0-60" 12.945715922966716 167.59156075815397 2 15 "15.14" [] [[21.0, 0.0]] 42 true 1 0 0.996971434396391215928732768981 ["ModularForm/GL2/Q/holomorphic/15/43/d/b", "ModularForm/GL2/Q/holomorphic/15/43/d/b/14/1"] "2-15-15.14-c44-0-27" 13.562064830578272 183.92960246880807 2 15 "15.14" [] [[22.0, 0.0]] 44 true 1 0 0.42756800497414526019181730130 ["ModularForm/GL2/Q/holomorphic/15/45/d/b", "ModularForm/GL2/Q/holomorphic/15/45/d/b/14/1"] "2-15-15.14-c44-0-55" 13.562064830578272 183.92960246880807 2 15 "15.14" [] [[22.0, 0.0]] 44 true 1 0 0.77086145132575867346568932160 ["ModularForm/GL2/Q/holomorphic/15/45/d/a", "ModularForm/GL2/Q/holomorphic/15/45/d/a/14/1"] "2-15-15.14-c46-0-40" 14.178418563038578 201.02755294871696 2 15 "15.14" [] [[23.0, 0.0]] 46 true 1 0 0.59748104789625716518936648689 ["ModularForm/GL2/Q/holomorphic/15/47/d/b", "ModularForm/GL2/Q/holomorphic/15/47/d/b/14/1"] "2-15-15.14-c46-0-46" 14.178418563038578 201.02755294871696 2 15 "15.14" [] [[23.0, 0.0]] 46 true 1 0 0.65625490511240463249017434021 ["ModularForm/GL2/Q/holomorphic/15/47/d/a", "ModularForm/GL2/Q/holomorphic/15/47/d/a/14/1"] "2-15-15.14-c48-0-43" 14.794776517929513 218.8854122154785 2 15 "15.14" [] [[24.0, 0.0]] 48 true 1 0 0.49230949354056365875808642086 ["ModularForm/GL2/Q/holomorphic/15/49/d/a", "ModularForm/GL2/Q/holomorphic/15/49/d/a/14/1"] "2-15-15.14-c48-0-86" 14.794776517929513 218.8854122154785 2 15 "15.14" [] [[24.0, 0.0]] 48 true 1 0 1.44466171960920538750170578162 ["ModularForm/GL2/Q/holomorphic/15/49/d/b", "ModularForm/GL2/Q/holomorphic/15/49/d/b/14/1"] "2-15-15.14-c50-0-32" 15.411138189090718 237.50318028325034 2 15 "15.14" [] [[25.0, 0.0]] 50 true 1 0 0.38970368841849635599121926181 ["ModularForm/GL2/Q/holomorphic/15/51/d/a", "ModularForm/GL2/Q/holomorphic/15/51/d/a/14/1"] "2-15-15.14-c50-0-60" 15.411138189090718 237.50318028325034 2 15 "15.14" [] [[25.0, 0.0]] 50 true 1 0 0.73041633432102229509316261508 ["ModularForm/GL2/Q/holomorphic/15/51/d/b", "ModularForm/GL2/Q/holomorphic/15/51/d/b/14/1"] # 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 $1$. #$\epsilon$ (root_number) -- # The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$. The sign appears as the 4th entry in the quadruple # known as the Selberg data. #$r$ (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).