# lfunc_search downloaded from the LMFDB on 05 April 2026. # Search link: https://www.lmfdb.org/L/1/2600/2600.253/r0-0 # Query "{'degree': 1, 'conductor': 2600, 'spectral_label': 'r0-0'}" returned 176 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. "1-2600-2600.1019-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1019" [[0, 0.0]] [] 0 true true false false 0.3067917604527497 0 0.920580520042 ["Character/Dirichlet/2600/1019"] "1-2600-2600.1059-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1059" [[0, 0.0]] [] 0 true true false false -0.3261455694823664 0 0.321162857112 ["Character/Dirichlet/2600/1059"] "1-2600-2600.1069-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1069" [[0, 0.0]] [] 0 true true false false 0.4693538090296167 0 1.79160913568 ["Character/Dirichlet/2600/1069"] "1-2600-2600.1077-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1077" [[0, 0.0]] [] 0 true true false false -0.1738544305176336 0 0.268435967198 ["Character/Dirichlet/2600/1077"] "1-2600-2600.1083-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1083" [[0, 0.0]] [] 0 true true false false 0.09537428472590004 0 0.93371860949 ["Character/Dirichlet/2600/1083"] "1-2600-2600.11-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.11" [[0, 0.0]] [] 0 true true false false -0.1738544305176336 0 0.873393261204 ["Character/Dirichlet/2600/11"] "1-2600-2600.1109-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1109" [[0, 0.0]] [] 0 true true false false 0.2653742847259 0 1.23579570119 ["Character/Dirichlet/2600/1109"] "1-2600-2600.1133-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1133" [[0, 0.0]] [] 0 true true false false 0.42256204857686697 0 0.161692901898 ["Character/Dirichlet/2600/1133"] "1-2600-2600.1139-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1139" [[0, 0.0]] [] 0 true true false false 0.2732082395472503 0 1.13469636181 ["Character/Dirichlet/2600/1139"] "1-2600-2600.1141-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1141" [[0, 0.0]] [] 0 true true false false -0.40537428472590004 0 1.57908012761 ["Character/Dirichlet/2600/1141"] "1-2600-2600.1147-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1147" [[0, 0.0]] [] 0 true true false false -0.30064619097038336 0 0.0623395655074 ["Character/Dirichlet/2600/1147"] "1-2600-2600.1187-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1187" [[0, 0.0]] [] 0 true true false false -0.23462571527409998 0 0.594724665479 ["Character/Dirichlet/2600/1187"] "1-2600-2600.1211-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1211" [[0, 0.0]] [] 0 true true false false 0.23256204857686696 0 1.15234120016 ["Character/Dirichlet/2600/1211"] "1-2600-2600.1221-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1221" [[0, 0.0]] [] 0 true true false false 0.24000000000000002 0 0.0222354190278 ["Character/Dirichlet/2600/1221"] "1-2600-2600.1237-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1237" [[0, 0.0]] [] 0 true true false false 0.09256204857686698 0 1.17822533989 ["Character/Dirichlet/2600/1237"] "1-2600-2600.1253-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1253" [[0, 0.0]] [] 0 true true false false 0.036791760452749706 0 1.13332006159 ["Character/Dirichlet/2600/1253"] "1-2600-2600.1259-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1259" [[0, 0.0]] [] 0 true true false false -0.3138544305176336 0 1.65911511997 ["Character/Dirichlet/2600/1259"] "1-2600-2600.1283-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1283" [[0, 0.0]] [] 0 true true false false 0.30064619097038336 0 1.17590326181 ["Character/Dirichlet/2600/1283"] "1-2600-2600.1309-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1309" [[0, 0.0]] [] 0 true true false false 0.47064619097038335 0 1.75407057334 ["Character/Dirichlet/2600/1309"] "1-2600-2600.1323-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1323" [[0, 0.0]] [] 0 true true false false 0.23462571527409998 0 0.852840926906 ["Character/Dirichlet/2600/1323"] "1-2600-2600.1331-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1331" [[0, 0.0]] [] 0 true true false false 0.3067917604527497 0 0.118688892053 ["Character/Dirichlet/2600/1331"] "1-2600-2600.1333-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1333" [[0, 0.0]] [] 0 true true false false -0.40256204857686706 0 0.00217552960037 ["Character/Dirichlet/2600/1333"] "1-2600-2600.1371-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1371" [[0, 0.0]] [] 0 true true false false -0.2661455694823664 0 0.595083645941 ["Character/Dirichlet/2600/1371"] "1-2600-2600.1373-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1373" [[0, 0.0]] [] 0 true true false false 0.2732082395472503 0 0.738398340747 ["Character/Dirichlet/2600/1373"] "1-2600-2600.1381-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1381" [[0, 0.0]] [] 0 true true false false 0.4693538090296167 0 0.130595651478 ["Character/Dirichlet/2600/1381"] "1-2600-2600.1387-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1387" [[0, 0.0]] [] 0 true true false false -0.029353809029616686 0 1.14285915033 ["Character/Dirichlet/2600/1387"] "1-2600-2600.1397-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1397" [[0, 0.0]] [] 0 true true false false -0.01614556948236639 0 0.847230844288 ["Character/Dirichlet/2600/1397"] "1-2600-2600.1403-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1403" [[0, 0.0]] [] 0 true true false false -0.01 0 0.702276990848 ["Character/Dirichlet/2600/1403"] "1-2600-2600.1411-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1411" [[0, 0.0]] [] 0 true true false false 0.40743795142313305 0 1.0742646472 ["Character/Dirichlet/2600/1411"] "1-2600-2600.1419-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1419" [[0, 0.0]] [] 0 true true false false -0.32743795142313303 0 0.503467568766 ["Character/Dirichlet/2600/1419"] "1-2600-2600.1421-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1421" [[0, 0.0]] [] 0 true true false false 0.3253742847259 0 1.11905944265 ["Character/Dirichlet/2600/1421"] "1-2600-2600.1427-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1427" [[0, 0.0]] [] 0 true true false false -0.2653742847259 0 0.56673429622 ["Character/Dirichlet/2600/1427"] "1-2600-2600.1429-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1429" [[0, 0.0]] [] 0 true true false false 0.24000000000000002 0 1.01027590424 ["Character/Dirichlet/2600/1429"] "1-2600-2600.1437-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1437" [[0, 0.0]] [] 0 true true false false 0.26743795142313304 0 0.882066824624 ["Character/Dirichlet/2600/1437"] "1-2600-2600.147-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.147" [[0, 0.0]] [] 0 true true false false 0.07537428472590003 0 0.748999149682 ["Character/Dirichlet/2600/147"] "1-2600-2600.1477-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1477" [[0, 0.0]] [] 0 true true false false -0.2267917604527497 0 0.764040160621 ["Character/Dirichlet/2600/1477"] "1-2600-2600.1539-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1539" [[0, 0.0]] [] 0 true true false false 0.3667917604527497 0 1.54987877098 ["Character/Dirichlet/2600/1539"] "1-2600-2600.1563-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1563" [[0, 0.0]] [] 0 true true false false -0.11064619097038332 0 0.872701551973 ["Character/Dirichlet/2600/1563"] "1-2600-2600.1571-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1571" [[0, 0.0]] [] 0 true true false false -0.24614556948236638 0 0.604285147206 ["Character/Dirichlet/2600/1571"] "1-2600-2600.1579-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1579" [[0, 0.0]] [] 0 true true false false -0.2661455694823664 0 0.20146422667 ["Character/Dirichlet/2600/1579"] "1-2600-2600.1589-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1589" [[0, 0.0]] [] 0 true true false false -0.47064619097038335 0 0.150324111635 ["Character/Dirichlet/2600/1589"] "1-2600-2600.1597-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1597" [[0, 0.0]] [] 0 true true false false 0.003854430517633615 0 0.948295250005 ["Character/Dirichlet/2600/1597"] "1-2600-2600.1603-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1603" [[0, 0.0]] [] 0 true true false false 0.07537428472590003 0 0.957444992646 ["Character/Dirichlet/2600/1603"] "1-2600-2600.1619-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1619" [[0, 0.0]] [] 0 true true false false -0.152562048576867 0 0.783809755846 ["Character/Dirichlet/2600/1619"] "1-2600-2600.1621-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1621" [[0, 0.0]] [] 0 true true false false -0.4693538090296167 0 1.15285295586 ["Character/Dirichlet/2600/1621"] "1-2600-2600.1629-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1629" [[0, 0.0]] [] 0 true true false false 0.3253742847259 0 1.36979977873 ["Character/Dirichlet/2600/1629"] "1-2600-2600.1653-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1653" [[0, 0.0]] [] 0 true true false false 0.40256204857686706 0 0.900400648812 ["Character/Dirichlet/2600/1653"] "1-2600-2600.1659-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1659" [[0, 0.0]] [] 0 true true false false -0.3667917604527497 0 0.622583604067 ["Character/Dirichlet/2600/1659"] "1-2600-2600.1661-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1661" [[0, 0.0]] [] 0 true true false false 0.23462571527409998 0 1.08297325354 ["Character/Dirichlet/2600/1661"] "1-2600-2600.1667-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1667" [[0, 0.0]] [] 0 true true false false -0.2806461909703833 0 0.519935972452 ["Character/Dirichlet/2600/1667"] "1-2600-2600.171-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.171" [[0, 0.0]] [] 0 true true false false 0.152562048576867 0 0.879495453272 ["Character/Dirichlet/2600/171"] "1-2600-2600.1731-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1731" [[0, 0.0]] [] 0 true true false false -0.32743795142313303 0 0.0240812481175 ["Character/Dirichlet/2600/1731"] "1-2600-2600.1741-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1741" [[0, 0.0]] [] 0 true true false false -0.32 0 0.416176580188 ["Character/Dirichlet/2600/1741"] "1-2600-2600.1773-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1773" [[0, 0.0]] [] 0 true true false false 0.3667917604527497 0 1.0099320335 ["Character/Dirichlet/2600/1773"] "1-2600-2600.1779-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1779" [[0, 0.0]] [] 0 true true false false -0.24614556948236638 0 0.320299179589 ["Character/Dirichlet/2600/1779"] "1-2600-2600.1803-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1803" [[0, 0.0]] [] 0 true true false false 0.2806461909703833 0 1.09996862185 ["Character/Dirichlet/2600/1803"] "1-2600-2600.181-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.181" [[0, 0.0]] [] 0 true true false false -0.24000000000000002 0 1.58176453876 ["Character/Dirichlet/2600/181"] "1-2600-2600.1813-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1813" [[0, 0.0]] [] 0 true true false false 0.1738544305176336 0 1.01259685717 ["Character/Dirichlet/2600/1813"] "1-2600-2600.1829-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1829" [[0, 0.0]] [] 0 true true false false -0.4693538090296167 0 0.492800969142 ["Character/Dirichlet/2600/1829"] "1-2600-2600.1853-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1853" [[0, 0.0]] [] 0 true true false false -0.42256204857686697 0 0.265055002556 ["Character/Dirichlet/2600/1853"] "1-2600-2600.1869-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1869" [[0, 0.0]] [] 0 true true false false -0.3253742847259 0 0.188062407823 ["Character/Dirichlet/2600/1869"] "1-2600-2600.1877-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1877" [[0, 0.0]] [] 0 true true false false -0.1332082395472503 0 0.705595763743 ["Character/Dirichlet/2600/1877"] "1-2600-2600.1891-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1891" [[0, 0.0]] [] 0 true true false false 0.1738544305176336 0 1.21617289522 ["Character/Dirichlet/2600/1891"] "1-2600-2600.19-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.19" [[0, 0.0]] [] 0 true true false false 0.24614556948236638 0 1.20284326282 ["Character/Dirichlet/2600/19"] "1-2600-2600.1917-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1917" [[0, 0.0]] [] 0 true true false false 0.003854430517633615 0 1.05295910933 ["Character/Dirichlet/2600/1917"] "1-2600-2600.1923-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1923" [[0, 0.0]] [] 0 true true false false 0.32 0 1.38847382614 ["Character/Dirichlet/2600/1923"] "1-2600-2600.1931-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1931" [[0, 0.0]] [] 0 true true false false -0.152562048576867 0 0.604245563812 ["Character/Dirichlet/2600/1931"] "1-2600-2600.1939-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1939" [[0, 0.0]] [] 0 true true false false -0.26743795142313304 0 0.313009332377 ["Character/Dirichlet/2600/1939"] "1-2600-2600.1941-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1941" [[0, 0.0]] [] 0 true true false false -0.23462571527409998 0 0.786698370238 ["Character/Dirichlet/2600/1941"] "1-2600-2600.1947-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1947" [[0, 0.0]] [] 0 true true false false -0.09537428472590004 0 0.390379888434 ["Character/Dirichlet/2600/1947"] "1-2600-2600.197-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.197" [[0, 0.0]] [] 0 true true false false 0.40256204857686706 0 1.56315413937 ["Character/Dirichlet/2600/197"] "1-2600-2600.1971-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1971" [[0, 0.0]] [] 0 true true false false -0.3067917604527497 0 1.92183492219 ["Character/Dirichlet/2600/1971"] "1-2600-2600.1997-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.1997" [[0, 0.0]] [] 0 true true false false -0.05679176045274971 0 0.733881767461 ["Character/Dirichlet/2600/1997"] "1-2600-2600.2013-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2013" [[0, 0.0]] [] 0 true true false false 0.1861455694823664 0 0.84401932314 ["Character/Dirichlet/2600/2013"] "1-2600-2600.2027-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2027" [[0, 0.0]] [] 0 true true false false -0.18 0 0.112924010765 ["Character/Dirichlet/2600/2027"] "1-2600-2600.2059-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2059" [[0, 0.0]] [] 0 true true false false -0.2732082395472503 0 0.431901821131 ["Character/Dirichlet/2600/2059"] "1-2600-2600.2083-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2083" [[0, 0.0]] [] 0 true true false false -0.2806461909703833 0 0.304780690975 ["Character/Dirichlet/2600/2083"] "1-2600-2600.2091-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2091" [[0, 0.0]] [] 0 true true false false 0.1861455694823664 0 1.14532363191 ["Character/Dirichlet/2600/2091"] "1-2600-2600.2109-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2109" [[0, 0.0]] [] 0 true true false false -0.11064619097038332 0 0.822694228771 ["Character/Dirichlet/2600/2109"] "1-2600-2600.2117-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2117" [[0, 0.0]] [] 0 true true false false 0.01614556948236639 0 0.671835418095 ["Character/Dirichlet/2600/2117"] "1-2600-2600.2123-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2123" [[0, 0.0]] [] 0 true true false false 0.40537428472590004 0 1.35500545583 ["Character/Dirichlet/2600/2123"] "1-2600-2600.213-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.213" [[0, 0.0]] [] 0 true true false false 0.2267917604527497 0 1.22878885737 ["Character/Dirichlet/2600/213"] "1-2600-2600.2139-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2139" [[0, 0.0]] [] 0 true true false false -0.09256204857686698 0 0.890986419317 ["Character/Dirichlet/2600/2139"] "1-2600-2600.2141-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2141" [[0, 0.0]] [] 0 true true false false -0.029353809029616686 0 0.510754286477 ["Character/Dirichlet/2600/2141"] "1-2600-2600.2173-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2173" [[0, 0.0]] [] 0 true true false false -0.26743795142313304 0 0.685746440846 ["Character/Dirichlet/2600/2173"] "1-2600-2600.2179-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2179" [[0, 0.0]] [] 0 true true false false -0.3067917604527497 0 0.540880248504 ["Character/Dirichlet/2600/2179"] "1-2600-2600.2181-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2181" [[0, 0.0]] [] 0 true true false false -0.3253742847259 0 0.435610885243 ["Character/Dirichlet/2600/2181"] "1-2600-2600.2187-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2187" [[0, 0.0]] [] 0 true true false false 0.3893538090296167 0 0.850614210804 ["Character/Dirichlet/2600/2187"] "1-2600-2600.219-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.219" [[0, 0.0]] [] 0 true true false false 0.2661455694823664 0 0.935181832876 ["Character/Dirichlet/2600/219"] "1-2600-2600.2227-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2227" [[0, 0.0]] [] 0 true true false false -0.09462571527409998 0 0.788756586621 ["Character/Dirichlet/2600/2227"] "1-2600-2600.2261-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2261" [[0, 0.0]] [] 0 true true false false 0.32 0 1.15178063202 ["Character/Dirichlet/2600/2261"] "1-2600-2600.2277-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2277" [[0, 0.0]] [] 0 true true false false 0.23256204857686696 0 1.04398141641 ["Character/Dirichlet/2600/2277"] "1-2600-2600.2323-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2323" [[0, 0.0]] [] 0 true true false false -0.3893538090296167 0 0.087694517908 ["Character/Dirichlet/2600/2323"] "1-2600-2600.2333-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2333" [[0, 0.0]] [] 0 true true false false 0.003854430517633615 0 0.49939441124 ["Character/Dirichlet/2600/2333"] "1-2600-2600.2363-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2363" [[0, 0.0]] [] 0 true true false false 0.09462571527409998 0 1.09056532657 ["Character/Dirichlet/2600/2363"] "1-2600-2600.2371-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2371" [[0, 0.0]] [] 0 true true false false -0.21320823954725032 0 0.32134294865 ["Character/Dirichlet/2600/2371"] "1-2600-2600.2373-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2373" [[0, 0.0]] [] 0 true true false false -0.09256204857686698 0 0.94415022589 ["Character/Dirichlet/2600/2373"] "1-2600-2600.2389-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2389" [[0, 0.0]] [] 0 true true false false -0.2653742847259 0 0.60354856151 ["Character/Dirichlet/2600/2389"] "1-2600-2600.2397-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2397" [[0, 0.0]] [] 0 true true false false 0.036791760452749706 0 0.423749698484 ["Character/Dirichlet/2600/2397"] "1-2600-2600.2411-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2411" [[0, 0.0]] [] 0 true true false false -0.1861455694823664 0 0.230688578867 ["Character/Dirichlet/2600/2411"] "1-2600-2600.2413-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2413" [[0, 0.0]] [] 0 true true false false 0.1332082395472503 0 0.734793856765 ["Character/Dirichlet/2600/2413"] "1-2600-2600.2421-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2421" [[0, 0.0]] [] 0 true true false false -0.050646190970383316 0 0.74912328522 ["Character/Dirichlet/2600/2421"] "1-2600-2600.2427-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2427" [[0, 0.0]] [] 0 true true false false 0.11064619097038332 0 1.15182040404 ["Character/Dirichlet/2600/2427"] "1-2600-2600.2437-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2437" [[0, 0.0]] [] 0 true true false false -0.3261455694823664 0 0.224751935397 ["Character/Dirichlet/2600/2437"] "1-2600-2600.2459-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2459" [[0, 0.0]] [] 0 true true false false 0.09256204857686698 0 0.835657698027 ["Character/Dirichlet/2600/2459"] "1-2600-2600.2461-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2461" [[0, 0.0]] [] 0 true true false false 0.40537428472590004 0 0.262510569854 ["Character/Dirichlet/2600/2461"] "1-2600-2600.2467-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2467" [[0, 0.0]] [] 0 true true false false -0.07537428472590003 0 1.17572438766 ["Character/Dirichlet/2600/2467"] "1-2600-2600.2469-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2469" [[0, 0.0]] [] 0 true true false false -0.24000000000000002 0 0.65428381527 ["Character/Dirichlet/2600/2469"] "1-2600-2600.2477-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2477" [[0, 0.0]] [] 0 true true false false 0.40743795142313305 0 1.01788829942 ["Character/Dirichlet/2600/2477"] "1-2600-2600.2491-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2491" [[0, 0.0]] [] 0 true true false false 0.1332082395472503 0 0.820632186074 ["Character/Dirichlet/2600/2491"] "1-2600-2600.2517-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2517" [[0, 0.0]] [] 0 true true false false -0.036791760452749706 0 0.840989885814 ["Character/Dirichlet/2600/2517"] "1-2600-2600.253-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.253" [[0, 0.0]] [] 0 true true false false -0.01614556948236639 0 0.704473676823 ["Character/Dirichlet/2600/253"] "1-2600-2600.2533-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2533" [[0, 0.0]] [] 0 true true false false 0.01614556948236639 0 1.20977698542 ["Character/Dirichlet/2600/2533"] "1-2600-2600.2547-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2547" [[0, 0.0]] [] 0 true true false false -0.01 0 1.00390448447 ["Character/Dirichlet/2600/2547"] "1-2600-2600.2579-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.2579" [[0, 0.0]] [] 0 true true false false -0.21320823954725032 0 0.578184962924 ["Character/Dirichlet/2600/2579"] "1-2600-2600.269-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.269" [[0, 0.0]] [] 0 true true false false 0.050646190970383316 0 0.827505506291 ["Character/Dirichlet/2600/269"] "1-2600-2600.283-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.283" [[0, 0.0]] [] 0 true true false false -0.07537428472590003 0 0.306742173403 ["Character/Dirichlet/2600/283"] "1-2600-2600.29-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.29" [[0, 0.0]] [] 0 true true false false -0.050646190970383316 0 0.593652825718 ["Character/Dirichlet/2600/29"] "1-2600-2600.291-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.291" [[0, 0.0]] [] 0 true true false false 0.2267917604527497 0 0.836253987781 ["Character/Dirichlet/2600/291"] "1-2600-2600.3-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.3" [[0, 0.0]] [] 0 true true false false -0.30064619097038336 0 0.7007999872 ["Character/Dirichlet/2600/3"] "1-2600-2600.309-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.309" [[0, 0.0]] [] 0 true true false false 0.09462571527409998 0 0.879141329159 ["Character/Dirichlet/2600/309"] "1-2600-2600.317-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.317" [[0, 0.0]] [] 0 true true false false 0.05679176045274971 0 0.899491959306 ["Character/Dirichlet/2600/317"] "1-2600-2600.331-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.331" [[0, 0.0]] [] 0 true true false false 0.24614556948236638 0 1.10442620506 ["Character/Dirichlet/2600/331"] "1-2600-2600.333-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.333" [[0, 0.0]] [] 0 true true false false -0.036791760452749706 0 0.447721278839 ["Character/Dirichlet/2600/333"] "1-2600-2600.341-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.341" [[0, 0.0]] [] 0 true true false false 0.3893538090296167 0 1.1028085948 ["Character/Dirichlet/2600/341"] "1-2600-2600.347-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.347" [[0, 0.0]] [] 0 true true false false 0.2806461909703833 0 1.15116244716 ["Character/Dirichlet/2600/347"] "1-2600-2600.363-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.363" [[0, 0.0]] [] 0 true true false false 0.18 0 0.910166387314 ["Character/Dirichlet/2600/363"] "1-2600-2600.37-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.37" [[0, 0.0]] [] 0 true true false false -0.3138544305176336 0 0.532858763227 ["Character/Dirichlet/2600/37"] "1-2600-2600.371-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.371" [[0, 0.0]] [] 0 true true false false 0.32743795142313303 0 1.31826762177 ["Character/Dirichlet/2600/371"] "1-2600-2600.379-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.379" [[0, 0.0]] [] 0 true true false false 0.152562048576867 0 1.33358605728 ["Character/Dirichlet/2600/379"] "1-2600-2600.381-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.381" [[0, 0.0]] [] 0 true true false false -0.15462571527409996 0 0.832749767682 ["Character/Dirichlet/2600/381"] "1-2600-2600.387-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.387" [[0, 0.0]] [] 0 true true false false -0.40537428472590004 0 0.175120344331 ["Character/Dirichlet/2600/387"] "1-2600-2600.389-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.389" [[0, 0.0]] [] 0 true true false false -0.18 0 0.668849111005 ["Character/Dirichlet/2600/389"] "1-2600-2600.397-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.397" [[0, 0.0]] [] 0 true true false false -0.42256204857686697 0 1.63536756173 ["Character/Dirichlet/2600/397"] "1-2600-2600.411-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.411" [[0, 0.0]] [] 0 true true false false -0.2267917604527497 0 0.382558301226 ["Character/Dirichlet/2600/411"] "1-2600-2600.437-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.437" [[0, 0.0]] [] 0 true true false false -0.3667917604527497 0 0.282905048996 ["Character/Dirichlet/2600/437"] "1-2600-2600.453-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.453" [[0, 0.0]] [] 0 true true false false 0.003854430517633615 0 0.487158166908 ["Character/Dirichlet/2600/453"] "1-2600-2600.467-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.467" [[0, 0.0]] [] 0 true true false false 0.01 0 0.691483945316 ["Character/Dirichlet/2600/467"] "1-2600-2600.523-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.523" [[0, 0.0]] [] 0 true true false false 0.029353809029616686 0 0.763393255972 ["Character/Dirichlet/2600/523"] "1-2600-2600.531-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.531" [[0, 0.0]] [] 0 true true false false 0.2661455694823664 0 1.18076036809 ["Character/Dirichlet/2600/531"] "1-2600-2600.539-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.539" [[0, 0.0]] [] 0 true true false false 0.3138544305176336 0 0.174602634596 ["Character/Dirichlet/2600/539"] "1-2600-2600.563-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.563" [[0, 0.0]] [] 0 true true false false 0.2653742847259 0 0.901548202691 ["Character/Dirichlet/2600/563"] "1-2600-2600.579-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.579" [[0, 0.0]] [] 0 true true false false 0.32743795142313303 0 1.14500202539 ["Character/Dirichlet/2600/579"] "1-2600-2600.581-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.581" [[0, 0.0]] [] 0 true true false false 0.050646190970383316 0 0.927910149255 ["Character/Dirichlet/2600/581"] "1-2600-2600.589-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.589" [[0, 0.0]] [] 0 true true false false -0.09462571527409998 0 0.221229489886 ["Character/Dirichlet/2600/589"] "1-2600-2600.59-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.59" [[0, 0.0]] [] 0 true true false false 0.26743795142313304 0 1.13497982039 ["Character/Dirichlet/2600/59"] "1-2600-2600.61-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.61" [[0, 0.0]] [] 0 true true false false -0.3893538090296167 0 0.314965941256 ["Character/Dirichlet/2600/61"] "1-2600-2600.613-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.613" [[0, 0.0]] [] 0 true true false false -0.40743795142313305 0 0.109655278717 ["Character/Dirichlet/2600/613"] "1-2600-2600.619-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.619" [[0, 0.0]] [] 0 true true false false 0.21320823954725032 0 0.878783661964 ["Character/Dirichlet/2600/619"] "1-2600-2600.621-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.621" [[0, 0.0]] [] 0 true true false false 0.15462571527409996 0 1.2196584972 ["Character/Dirichlet/2600/621"] "1-2600-2600.627-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.627" [[0, 0.0]] [] 0 true true false false -0.47064619097038335 0 0.0498104643918 ["Character/Dirichlet/2600/627"] "1-2600-2600.667-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.667" [[0, 0.0]] [] 0 true true false false 0.09537428472590004 0 0.539445736702 ["Character/Dirichlet/2600/667"] "1-2600-2600.69-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.69" [[0, 0.0]] [] 0 true true false false -0.15462571527409996 0 0.357685545981 ["Character/Dirichlet/2600/69"] "1-2600-2600.691-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.691" [[0, 0.0]] [] 0 true true false false -0.40743795142313305 0 0.382905997106 ["Character/Dirichlet/2600/691"] "1-2600-2600.717-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.717" [[0, 0.0]] [] 0 true true false false 0.42256204857686697 0 1.03903613323 ["Character/Dirichlet/2600/717"] "1-2600-2600.733-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.733" [[0, 0.0]] [] 0 true true false false 0.05679176045274971 0 0.945847578958 ["Character/Dirichlet/2600/733"] "1-2600-2600.739-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.739" [[0, 0.0]] [] 0 true true false false 0.3261455694823664 0 0.943626109938 ["Character/Dirichlet/2600/739"] "1-2600-2600.763-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.763" [[0, 0.0]] [] 0 true true false false 0.47064619097038335 0 0.966341376883 ["Character/Dirichlet/2600/763"] "1-2600-2600.773-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.773" [[0, 0.0]] [] 0 true true false false 0.3138544305176336 0 1.50196086661 ["Character/Dirichlet/2600/773"] "1-2600-2600.789-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.789" [[0, 0.0]] [] 0 true true false false 0.11064619097038332 0 0.844604797963 ["Character/Dirichlet/2600/789"] "1-2600-2600.803-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.803" [[0, 0.0]] [] 0 true true false false -0.09537428472590004 0 0.559365124603 ["Character/Dirichlet/2600/803"] "1-2600-2600.811-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.811" [[0, 0.0]] [] 0 true true false false -0.1332082395472503 0 0.829444628374 ["Character/Dirichlet/2600/811"] "1-2600-2600.813-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.813" [[0, 0.0]] [] 0 true true false false -0.23256204857686696 0 0.764556254958 ["Character/Dirichlet/2600/813"] "1-2600-2600.829-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.829" [[0, 0.0]] [] 0 true true false false 0.15462571527409996 0 0.95848245808 ["Character/Dirichlet/2600/829"] "1-2600-2600.837-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.837" [[0, 0.0]] [] 0 true true false false -0.2732082395472503 0 0.324317616078 ["Character/Dirichlet/2600/837"] "1-2600-2600.853-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.853" [[0, 0.0]] [] 0 true true false false -0.05679176045274971 0 0.923455825387 ["Character/Dirichlet/2600/853"] "1-2600-2600.861-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.861" [[0, 0.0]] [] 0 true true false false 0.029353809029616686 0 0.611677210714 ["Character/Dirichlet/2600/861"] "1-2600-2600.867-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.867" [[0, 0.0]] [] 0 true true false false 0.30064619097038336 0 1.18831106688 ["Character/Dirichlet/2600/867"] "1-2600-2600.877-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.877" [[0, 0.0]] [] 0 true true false false -0.1861455694823664 0 0.453331568736 ["Character/Dirichlet/2600/877"] "1-2600-2600.883-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.883" [[0, 0.0]] [] 0 true true false false 0.01 0 1.06984971566 ["Character/Dirichlet/2600/883"] "1-2600-2600.891-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.891" [[0, 0.0]] [] 0 true true false false -0.23256204857686696 0 0.348681962136 ["Character/Dirichlet/2600/891"] "1-2600-2600.909-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.909" [[0, 0.0]] [] 0 true true false false 0.18 0 0.937152475053 ["Character/Dirichlet/2600/909"] "1-2600-2600.917-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.917" [[0, 0.0]] [] 0 true true false false -0.40256204857686706 0 0.117173542649 ["Character/Dirichlet/2600/917"] "1-2600-2600.931-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.931" [[0, 0.0]] [] 0 true true false false 0.21320823954725032 0 1.1930134356 ["Character/Dirichlet/2600/931"] "1-2600-2600.973-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.973" [[0, 0.0]] [] 0 true true false false 0.3261455694823664 0 1.24401513309 ["Character/Dirichlet/2600/973"] "1-2600-2600.987-r0-0-0" 12.07434923837313 12.07434923837313 1 2600 "2600.987" [[0, 0.0]] [] 0 true true false false -0.32 0 0.0353966388677 ["Character/Dirichlet/2600/987"] # 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).