# lfunc_search downloaded from the LMFDB on 31 May 2026. # Search link: https://www.lmfdb.org/L/1/3648/3648.1229/r0-0 # Query "{'degree': 1, 'conductor': 3648, 'spectral_label': 'r0-0'}" returned 136 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. "1-3648-3648.1043-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1043" [[0, 0.0]] [] 0 true true false false 0.09297056790153949 0 0.953746668168 ["Character/Dirichlet/3648/1043"] "1-3648-3648.1085-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1085" [[0, 0.0]] [] 0 true true false false 0.3834655737749853 0 1.37530359327 ["Character/Dirichlet/3648/1085"] "1-3648-3648.11-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.11" [[0, 0.0]] [] 0 true true false false -0.2421650226355283 0 0.403823097956 ["Character/Dirichlet/3648/11"] "1-3648-3648.1133-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1133" [[0, 0.0]] [] 0 true true false false -0.19787171193772324 0 0.224648851527 ["Character/Dirichlet/3648/1133"] "1-3648-3648.1163-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1163" [[0, 0.0]] [] 0 true true false false -0.36557774721236413 0 0.109581175673 ["Character/Dirichlet/3648/1163"] "1-3648-3648.1181-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1181" [[0, 0.0]] [] 0 true true false false 0.4705899920701146 0 0.0645720310005 ["Character/Dirichlet/3648/1181"] "1-3648-3648.1187-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1187" [[0, 0.0]] [] 0 true true false false 0.0007794320984605124 0 0.599169600988 ["Character/Dirichlet/3648/1187"] "1-3648-3648.1205-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1205" [[0, 0.0]] [] 0 true true false false 0.07287171193772322 0 0.76504069011 ["Character/Dirichlet/3648/1205"] "1-3648-3648.1229-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1229" [[0, 0.0]] [] 0 true true false false -0.37683999207011465 0 0.249089035841 ["Character/Dirichlet/3648/1229"] "1-3648-3648.1253-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1253" [[0, 0.0]] [] 0 true true false false 0.265625 0 1.18589905635 ["Character/Dirichlet/3648/1253"] "1-3648-3648.1259-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1259" [[0, 0.0]] [] 0 true true false false 0.1468277472123641 0 0.908050006531 ["Character/Dirichlet/3648/1259"] "1-3648-3648.131-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.131" [[0, 0.0]] [] 0 true true false false 0.2179705679015395 0 0.678352447035 ["Character/Dirichlet/3648/131"] "1-3648-3648.1325-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1325" [[0, 0.0]] [] 0 true true false false -0.13535052057404337 0 0.386790884588 ["Character/Dirichlet/3648/1325"] "1-3648-3648.1355-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1355" [[0, 0.0]] [] 0 true true false false 0.054880004416098555 0 0.672285848087 ["Character/Dirichlet/3648/1355"] "1-3648-3648.1379-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1379" [[0, 0.0]] [] 0 true true false false -0.02341502263552828 0 0.605540772108 ["Character/Dirichlet/3648/1379"] "1-3648-3648.1397-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1397" [[0, 0.0]] [] 0 true true false false -0.3834655737749853 0 0.335063831272 ["Character/Dirichlet/3648/1397"] "1-3648-3648.1403-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1403" [[0, 0.0]] [] 0 true true false false 0.3513699955839015 0 1.27513467526 ["Character/Dirichlet/3648/1403"] "1-3648-3648.1421-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1421" [[0, 0.0]] [] 0 true true false false 0.10410052057404334 0 0.865596536363 ["Character/Dirichlet/3648/1421"] "1-3648-3648.1451-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1451" [[0, 0.0]] [] 0 true true false false 0.02341502263552828 0 0.888652072273 ["Character/Dirichlet/3648/1451"] "1-3648-3648.1499-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1499" [[0, 0.0]] [] 0 true true false false 0.1242205679015395 0 0.887697657496 ["Character/Dirichlet/3648/1499"] "1-3648-3648.1541-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1541" [[0, 0.0]] [] 0 true true false false 0.2897155737749853 0 0.991178312142 ["Character/Dirichlet/3648/1541"] "1-3648-3648.1589-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1589" [[0, 0.0]] [] 0 true true false false 0.2083782880622768 0 1.14165846918 ["Character/Dirichlet/3648/1589"] "1-3648-3648.1619-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1619" [[0, 0.0]] [] 0 true true false false -0.27182774721236413 0 0.415227683281 ["Character/Dirichlet/3648/1619"] "1-3648-3648.1637-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1637" [[0, 0.0]] [] 0 true true false false -0.12316000792988539 0 0.756820709982 ["Character/Dirichlet/3648/1637"] "1-3648-3648.1643-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1643" [[0, 0.0]] [] 0 true true false false -0.2179705679015395 0 0.0374206417922 ["Character/Dirichlet/3648/1643"] "1-3648-3648.1661-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1661" [[0, 0.0]] [] 0 true true false false -0.2083782880622768 0 0.266353496577 ["Character/Dirichlet/3648/1661"] "1-3648-3648.1685-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1685" [[0, 0.0]] [] 0 true true false false -0.4705899920701146 0 0.360249939421 ["Character/Dirichlet/3648/1685"] "1-3648-3648.1709-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1709" [[0, 0.0]] [] 0 true true false false -0.265625 0 0.436570772809 ["Character/Dirichlet/3648/1709"] "1-3648-3648.1715-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1715" [[0, 0.0]] [] 0 true true false false -0.25942225278763587 0 0.25813293948 ["Character/Dirichlet/3648/1715"] "1-3648-3648.173-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.173" [[0, 0.0]] [] 0 true true false false 0.2584655737749853 0 1.11554012554 ["Character/Dirichlet/3648/173"] "1-3648-3648.1781-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1781" [[0, 0.0]] [] 0 true true false false 0.27089947942595666 0 1.01517454704 ["Character/Dirichlet/3648/1781"] "1-3648-3648.1811-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1811" [[0, 0.0]] [] 0 true true false false 0.14863000441609855 0 0.406350352804 ["Character/Dirichlet/3648/1811"] "1-3648-3648.1835-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1835" [[0, 0.0]] [] 0 true true false false -0.2421650226355283 0 0.531927246507 ["Character/Dirichlet/3648/1835"] "1-3648-3648.1853-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1853" [[0, 0.0]] [] 0 true true false false 0.3352844262250147 0 0.0192140031573 ["Character/Dirichlet/3648/1853"] "1-3648-3648.1859-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1859" [[0, 0.0]] [] 0 true true false false -0.054880004416098555 0 0.628624353675 ["Character/Dirichlet/3648/1859"] "1-3648-3648.1877-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1877" [[0, 0.0]] [] 0 true true false false 0.010350520574043348 0 0.896108809439 ["Character/Dirichlet/3648/1877"] "1-3648-3648.1907-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1907" [[0, 0.0]] [] 0 true true false false -0.3828349773644717 0 0.5755544556 ["Character/Dirichlet/3648/1907"] "1-3648-3648.1955-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1955" [[0, 0.0]] [] 0 true true false false 0.2179705679015395 0 1.26061703101 ["Character/Dirichlet/3648/1955"] "1-3648-3648.1997-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.1997" [[0, 0.0]] [] 0 true true false false -0.24153442622501473 0 0.294772712671 ["Character/Dirichlet/3648/1997"] "1-3648-3648.2045-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2045" [[0, 0.0]] [] 0 true true false false -0.07287171193772322 0 0.849394254522 ["Character/Dirichlet/3648/2045"] "1-3648-3648.2075-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2075" [[0, 0.0]] [] 0 true true false false -0.2405777472123641 0 0.613508950443 ["Character/Dirichlet/3648/2075"] "1-3648-3648.2093-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2093" [[0, 0.0]] [] 0 true true false false 0.3455899920701146 0 1.02307748724 ["Character/Dirichlet/3648/2093"] "1-3648-3648.2099-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2099" [[0, 0.0]] [] 0 true true false false 0.3757794320984605 0 1.06154681477 ["Character/Dirichlet/3648/2099"] "1-3648-3648.2117-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2117" [[0, 0.0]] [] 0 true true false false -0.3021282880622768 0 0.360694617926 ["Character/Dirichlet/3648/2117"] "1-3648-3648.2141-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2141" [[0, 0.0]] [] 0 true true false false 0.24816000792988538 0 0.680191338684 ["Character/Dirichlet/3648/2141"] "1-3648-3648.2165-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2165" [[0, 0.0]] [] 0 true true false false 0.14062500000000003 0 1.27636950278 ["Character/Dirichlet/3648/2165"] "1-3648-3648.2171-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2171" [[0, 0.0]] [] 0 true true false false -0.2281722527876359 0 0.421626501474 ["Character/Dirichlet/3648/2171"] "1-3648-3648.221-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.221" [[0, 0.0]] [] 0 true true false false -0.07287171193772322 0 0.736142245835 ["Character/Dirichlet/3648/221"] "1-3648-3648.2237-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2237" [[0, 0.0]] [] 0 true true false false -0.010350520574043348 0 0.873873046519 ["Character/Dirichlet/3648/2237"] "1-3648-3648.2267-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2267" [[0, 0.0]] [] 0 true true false false 0.17988000441609853 0 0.927718939687 ["Character/Dirichlet/3648/2267"] "1-3648-3648.2291-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2291" [[0, 0.0]] [] 0 true true false false 0.35158497736447175 0 1.28996969869 ["Character/Dirichlet/3648/2291"] "1-3648-3648.2309-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2309" [[0, 0.0]] [] 0 true true false false 0.24153442622501473 0 0.930536178866 ["Character/Dirichlet/3648/2309"] "1-3648-3648.2315-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2315" [[0, 0.0]] [] 0 true true false false -0.2736300044160986 0 0.259919392314 ["Character/Dirichlet/3648/2315"] "1-3648-3648.2333-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2333" [[0, 0.0]] [] 0 true true false false -0.27089947942595666 0 0.30004750182 ["Character/Dirichlet/3648/2333"] "1-3648-3648.2363-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2363" [[0, 0.0]] [] 0 true true false false -0.35158497736447175 0 0.251350026605 ["Character/Dirichlet/3648/2363"] "1-3648-3648.2411-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2411" [[0, 0.0]] [] 0 true true false false 0.0007794320984605124 0 0.815353752283 ["Character/Dirichlet/3648/2411"] "1-3648-3648.2453-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2453" [[0, 0.0]] [] 0 true true false false -0.3352844262250147 0 1.579661758 ["Character/Dirichlet/3648/2453"] "1-3648-3648.2501-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2501" [[0, 0.0]] [] 0 true true false false -0.16662171193772324 0 0.768417169944 ["Character/Dirichlet/3648/2501"] "1-3648-3648.251-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.251" [[0, 0.0]] [] 0 true true false false 0.25942225278763587 0 0.832111584927 ["Character/Dirichlet/3648/251"] "1-3648-3648.2531-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2531" [[0, 0.0]] [] 0 true true false false -0.1468277472123641 0 0.879315728523 ["Character/Dirichlet/3648/2531"] "1-3648-3648.2549-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2549" [[0, 0.0]] [] 0 true true false false -0.24816000792988538 0 0.042505745124 ["Character/Dirichlet/3648/2549"] "1-3648-3648.2555-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2555" [[0, 0.0]] [] 0 true true false false 0.40702943209846054 0 0.19780382238 ["Character/Dirichlet/3648/2555"] "1-3648-3648.2573-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2573" [[0, 0.0]] [] 0 true true false false 0.16662171193772324 0 0.944885957993 ["Character/Dirichlet/3648/2573"] "1-3648-3648.2597-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2597" [[0, 0.0]] [] 0 true true false false -0.3455899920701146 0 0.198519224488 ["Character/Dirichlet/3648/2597"] "1-3648-3648.2621-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2621" [[0, 0.0]] [] 0 true true false false -0.14062500000000003 0 0.633946121095 ["Character/Dirichlet/3648/2621"] "1-3648-3648.2627-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2627" [[0, 0.0]] [] 0 true true false false 0.36557774721236413 0 0.00533883546165 ["Character/Dirichlet/3648/2627"] "1-3648-3648.269-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.269" [[0, 0.0]] [] 0 true true false false -0.15441000792988538 0 0.867494574383 ["Character/Dirichlet/3648/269"] "1-3648-3648.2693-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2693" [[0, 0.0]] [] 0 true true false false -0.10410052057404334 0 0.845483422419 ["Character/Dirichlet/3648/2693"] "1-3648-3648.2723-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2723" [[0, 0.0]] [] 0 true true false false 0.2736300044160986 0 1.1566540171 ["Character/Dirichlet/3648/2723"] "1-3648-3648.2747-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2747" [[0, 0.0]] [] 0 true true false false 0.3828349773644717 0 1.3501834771 ["Character/Dirichlet/3648/2747"] "1-3648-3648.275-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.275" [[0, 0.0]] [] 0 true true false false -0.1242205679015395 0 0.548174495448 ["Character/Dirichlet/3648/275"] "1-3648-3648.2765-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2765" [[0, 0.0]] [] 0 true true false false -0.2897155737749853 0 0.327354311359 ["Character/Dirichlet/3648/2765"] "1-3648-3648.2771-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2771" [[0, 0.0]] [] 0 true true false false -0.17988000441609853 0 0.423367953234 ["Character/Dirichlet/3648/2771"] "1-3648-3648.2789-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2789" [[0, 0.0]] [] 0 true true false false 0.13535052057404337 0 0.799151260813 ["Character/Dirichlet/3648/2789"] "1-3648-3648.2819-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2819" [[0, 0.0]] [] 0 true true false false 0.2421650226355283 0 1.15506699829 ["Character/Dirichlet/3648/2819"] "1-3648-3648.2867-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2867" [[0, 0.0]] [] 0 true true false false -0.40702943209846054 0 1.55800780516 ["Character/Dirichlet/3648/2867"] "1-3648-3648.29-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.29" [[0, 0.0]] [] 0 true true false false 0.3352844262250147 0 1.05344599943 ["Character/Dirichlet/3648/29"] "1-3648-3648.2909-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2909" [[0, 0.0]] [] 0 true true false false 0.3834655737749853 0 1.29101154998 ["Character/Dirichlet/3648/2909"] "1-3648-3648.293-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.293" [[0, 0.0]] [] 0 true true false false 0.19787171193772324 0 0.888427782884 ["Character/Dirichlet/3648/293"] "1-3648-3648.2957-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2957" [[0, 0.0]] [] 0 true true false false 0.3021282880622768 0 1.23443353559 ["Character/Dirichlet/3648/2957"] "1-3648-3648.2987-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.2987" [[0, 0.0]] [] 0 true true false false -0.36557774721236413 0 1.70675249087 ["Character/Dirichlet/3648/2987"] "1-3648-3648.3005-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3005" [[0, 0.0]] [] 0 true true false false 0.4705899920701146 0 1.43578099251 ["Character/Dirichlet/3648/3005"] "1-3648-3648.3011-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3011" [[0, 0.0]] [] 0 true true false false 0.0007794320984605124 0 0.915315478097 ["Character/Dirichlet/3648/3011"] "1-3648-3648.3029-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3029" [[0, 0.0]] [] 0 true true false false 0.07287171193772322 0 1.0701820978 ["Character/Dirichlet/3648/3029"] "1-3648-3648.3053-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3053" [[0, 0.0]] [] 0 true true false false 0.12316000792988539 0 0.913625385234 ["Character/Dirichlet/3648/3053"] "1-3648-3648.3077-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3077" [[0, 0.0]] [] 0 true true false false -0.23437500000000003 0 0.393269881348 ["Character/Dirichlet/3648/3077"] "1-3648-3648.3083-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3083" [[0, 0.0]] [] 0 true true false false 0.1468277472123641 0 1.4009481277 ["Character/Dirichlet/3648/3083"] "1-3648-3648.3149-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3149" [[0, 0.0]] [] 0 true true false false 0.36464947942595666 0 0.0926424189164 ["Character/Dirichlet/3648/3149"] "1-3648-3648.317-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.317" [[0, 0.0]] [] 0 true true false false 0.24816000792988538 0 0.946813051194 ["Character/Dirichlet/3648/317"] "1-3648-3648.3179-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3179" [[0, 0.0]] [] 0 true true false false 0.054880004416098555 0 0.807041462625 ["Character/Dirichlet/3648/3179"] "1-3648-3648.3203-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3203" [[0, 0.0]] [] 0 true true false false -0.02341502263552828 0 0.497020212947 ["Character/Dirichlet/3648/3203"] "1-3648-3648.3221-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3221" [[0, 0.0]] [] 0 true true false false -0.3834655737749853 0 0.373295616262 ["Character/Dirichlet/3648/3221"] "1-3648-3648.3227-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3227" [[0, 0.0]] [] 0 true true false false -0.14863000441609855 0 0.191148116503 ["Character/Dirichlet/3648/3227"] "1-3648-3648.3245-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3245" [[0, 0.0]] [] 0 true true false false 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["Character/Dirichlet/3648/3413"] "1-3648-3648.3443-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3443" [[0, 0.0]] [] 0 true true false false 0.2281722527876359 0 0.874028126053 ["Character/Dirichlet/3648/3443"] "1-3648-3648.3461-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3461" [[0, 0.0]] [] 0 true true false false 0.37683999207011465 0 1.25110095372 ["Character/Dirichlet/3648/3461"] "1-3648-3648.3467-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3467" [[0, 0.0]] [] 0 true true false false -0.2179705679015395 0 0.532714685207 ["Character/Dirichlet/3648/3467"] "1-3648-3648.347-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.347" [[0, 0.0]] [] 0 true true false false 0.27182774721236413 0 0.836399076231 ["Character/Dirichlet/3648/347"] "1-3648-3648.3485-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3485" [[0, 0.0]] [] 0 true true false false -0.2083782880622768 0 0.69638454749 ["Character/Dirichlet/3648/3485"] "1-3648-3648.35-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.35" [[0, 0.0]] [] 0 true true false false -0.054880004416098555 0 0.87131910351 ["Character/Dirichlet/3648/35"] "1-3648-3648.3509-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3509" [[0, 0.0]] [] 0 true true false false -0.4705899920701146 0 1.42354447141 ["Character/Dirichlet/3648/3509"] "1-3648-3648.3533-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3533" [[0, 0.0]] [] 0 true true false false 0.23437500000000003 0 0.726141281267 ["Character/Dirichlet/3648/3533"] "1-3648-3648.3539-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3539" [[0, 0.0]] [] 0 true true false false 0.2405777472123641 0 1.13994580282 ["Character/Dirichlet/3648/3539"] "1-3648-3648.3605-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3605" [[0, 0.0]] [] 0 true true false false 0.27089947942595666 0 0.915737657809 ["Character/Dirichlet/3648/3605"] "1-3648-3648.3635-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.3635" [[0, 0.0]] [] 0 true true false false -0.3513699955839015 0 0.400988945811 ["Character/Dirichlet/3648/3635"] "1-3648-3648.413-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.413" [[0, 0.0]] [] 0 true true false false -0.010350520574043348 0 0.449502546843 ["Character/Dirichlet/3648/413"] "1-3648-3648.443-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.443" [[0, 0.0]] [] 0 true true false false -0.32011999558390153 0 0.475460609055 ["Character/Dirichlet/3648/443"] "1-3648-3648.467-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.467" [[0, 0.0]] [] 0 true true false false -0.14841502263552828 0 0.617078497155 ["Character/Dirichlet/3648/467"] "1-3648-3648.485-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.485" [[0, 0.0]] [] 0 true true false false -0.2584655737749853 0 0.157096558429 ["Character/Dirichlet/3648/485"] "1-3648-3648.491-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.491" [[0, 0.0]] [] 0 true true false false -0.2736300044160986 0 0.515982885752 ["Character/Dirichlet/3648/491"] "1-3648-3648.509-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.509" [[0, 0.0]] [] 0 true true false false -0.27089947942595666 0 0.409136920677 ["Character/Dirichlet/3648/509"] "1-3648-3648.53-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.53" [[0, 0.0]] [] 0 true true false false 0.010350520574043348 0 0.678723963169 ["Character/Dirichlet/3648/53"] "1-3648-3648.539-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.539" [[0, 0.0]] [] 0 true true false false 0.14841502263552828 0 0.825535293138 ["Character/Dirichlet/3648/539"] "1-3648-3648.587-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.587" [[0, 0.0]] [] 0 true true false false 0.0007794320984605124 0 0.780830977908 ["Character/Dirichlet/3648/587"] "1-3648-3648.629-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.629" [[0, 0.0]] [] 0 true true false false -0.3352844262250147 0 0.477228431566 ["Character/Dirichlet/3648/629"] "1-3648-3648.677-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.677" [[0, 0.0]] [] 0 true true false false 0.33337828806227676 0 0.798308458685 ["Character/Dirichlet/3648/677"] "1-3648-3648.707-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.707" [[0, 0.0]] [] 0 true true false false -0.1468277472123641 0 0.384272881691 ["Character/Dirichlet/3648/707"] "1-3648-3648.725-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.725" [[0, 0.0]] [] 0 true true false false -0.24816000792988538 0 0.386673294575 ["Character/Dirichlet/3648/725"] "1-3648-3648.731-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.731" [[0, 0.0]] [] 0 true true false false -0.09297056790153949 0 0.617721779404 ["Character/Dirichlet/3648/731"] "1-3648-3648.749-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.749" [[0, 0.0]] [] 0 true true false false -0.33337828806227676 0 0.151226277309 ["Character/Dirichlet/3648/749"] "1-3648-3648.773-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.773" [[0, 0.0]] [] 0 true true false false 0.15441000792988538 0 1.06227437442 ["Character/Dirichlet/3648/773"] "1-3648-3648.797-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.797" [[0, 0.0]] [] 0 true true false false -0.14062500000000003 0 0.660697466082 ["Character/Dirichlet/3648/797"] "1-3648-3648.803-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.803" [[0, 0.0]] [] 0 true true false false 0.36557774721236413 0 0.97155184472 ["Character/Dirichlet/3648/803"] "1-3648-3648.83-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.83" [[0, 0.0]] [] 0 true true false false 0.11716502263552828 0 0.932994390037 ["Character/Dirichlet/3648/83"] "1-3648-3648.869-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.869" [[0, 0.0]] [] 0 true true false false 0.3958994794259567 0 1.48100771579 ["Character/Dirichlet/3648/869"] "1-3648-3648.899-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.899" [[0, 0.0]] [] 0 true true false false 0.2736300044160986 0 1.17996088889 ["Character/Dirichlet/3648/899"] "1-3648-3648.923-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.923" [[0, 0.0]] [] 0 true true false false -0.11716502263552828 0 0.886636030483 ["Character/Dirichlet/3648/923"] "1-3648-3648.941-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.941" [[0, 0.0]] [] 0 true true false false 0.21028442622501473 0 1.35366241091 ["Character/Dirichlet/3648/941"] "1-3648-3648.947-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.947" [[0, 0.0]] [] 0 true true false false 0.32011999558390153 0 1.38078000026 ["Character/Dirichlet/3648/947"] "1-3648-3648.965-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.965" [[0, 0.0]] [] 0 true true false false -0.36464947942595666 0 1.58240974471 ["Character/Dirichlet/3648/965"] "1-3648-3648.995-r0-0-0" 16.941240777532762 16.941240777532762 1 3648 "3648.995" [[0, 0.0]] [] 0 true true false false 0.2421650226355283 0 0.831501265917 ["Character/Dirichlet/3648/995"] # 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).