# lfunc_search downloaded from the LMFDB on 25 June 2026. # Search link: https://www.lmfdb.org/L/2/15/5.4/c17-0 # Query "{'degree': 2, 'conductor': 15, 'spectral_label': 'c17-0'}" returned 92 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-15-1.1-c17-0-0" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.0 0 0.36325077466978849698547560827 ["ModularForm/GL2/Q/holomorphic/15/18/a/d/1/2"] "2-15-1.1-c17-0-1" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.0 0 0.65471667732610118785935976027 ["ModularForm/GL2/Q/holomorphic/15/18/a/c/1/2"] "2-15-1.1-c17-0-10" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.5 1 2.19054759808334207002216786539 ["ModularForm/GL2/Q/holomorphic/15/18/a/b/1/3"] "2-15-1.1-c17-0-11" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.5 1 2.47722345968304888298274321211 ["ModularForm/GL2/Q/holomorphic/15/18/a/a/1/2"] "2-15-1.1-c17-0-2" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.0 0 0.802194415274953505666270138771 ["ModularForm/GL2/Q/holomorphic/15/18/a/c/1/1"] "2-15-1.1-c17-0-3" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.0 0 0.900611194307495217666167412874 ["ModularForm/GL2/Q/holomorphic/15/18/a/d/1/3"] "2-15-1.1-c17-0-4" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.0 0 0.955166660030715406991007639625 ["ModularForm/GL2/Q/holomorphic/15/18/a/d/1/1"] "2-15-1.1-c17-0-5" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.5 1 1.11212651359037415654593838235 ["ModularForm/GL2/Q/holomorphic/15/18/a/b/1/1"] "2-15-1.1-c17-0-6" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.0 0 1.23511750530014350801509419154 ["ModularForm/GL2/Q/holomorphic/15/18/a/c/1/3"] "2-15-1.1-c17-0-7" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.5 1 1.34401625379564660685743950964 ["ModularForm/GL2/Q/holomorphic/15/18/a/b/1/2"] "2-15-1.1-c17-0-8" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.0 0 1.49889297665907234001738396594 ["ModularForm/GL2/Q/holomorphic/15/18/a/d/1/4"] "2-15-1.1-c17-0-9" 5.242452966091729 27.48331310168397 2 15 "1.1" [] [[8.5, 0.0]] 17 true true false true 0.5 1 1.61711192957939357791759863108 ["ModularForm/GL2/Q/holomorphic/15/18/a/a/1/1"] "2-15-15.2-c17-0-0" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.16198503603171313 0 0.03800480080727800188176923531 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/14"] "2-15-15.2-c17-0-1" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.3133382581878896 0 0.12377275026493627878428471849 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/4"] "2-15-15.2-c17-0-10" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.14735168033145762 0 0.49753387810296998068973179701 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/23"] "2-15-15.2-c17-0-11" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.01456656305447067 0 0.64187840628601523469956004188 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/15"] "2-15-15.2-c17-0-12" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.3256307089094556 0 0.74217827511531556887607870086 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/9"] "2-15-15.2-c17-0-13" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.12255244519713235 0 0.934118191989323422252582039328 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/25"] "2-15-15.2-c17-0-14" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.11811445718746881 0 0.996651095938417452476368731935 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/26"] "2-15-15.2-c17-0-15" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.20639509639586523 0 1.01860311587582110959750493173 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/18"] "2-15-15.2-c17-0-16" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.126783254034373 0 1.08356896052463926758951116203 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/3"] "2-15-15.2-c17-0-17" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.21325539906305174 0 1.08426447581533036024635314473 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/11"] "2-15-15.2-c17-0-18" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.22559940967143932 0 1.11592906073597592553872368791 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/5"] "2-15-15.2-c17-0-19" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.05798834814654406 0 1.23065841874960724557340730422 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/30"] "2-15-15.2-c17-0-2" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.4874617875470529 0 0.17614459198546342171739172526 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/17"] "2-15-15.2-c17-0-20" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.08835387578163467 0 1.25000599218093175719664102647 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/13"] "2-15-15.2-c17-0-21" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.16940467305967666 0 1.30993924080149905784776400585 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/2"] "2-15-15.2-c17-0-22" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.2986450423015509 0 1.37947319492907568489093309875 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/27"] "2-15-15.2-c17-0-23" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.039416581977112564 0 1.38501650011919496326668079604 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/12"] "2-15-15.2-c17-0-24" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.3645303374262022 0 1.66574302430196069187231645236 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/22"] "2-15-15.2-c17-0-25" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.24613779900317928 0 1.68939363122647288669345682798 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/24"] "2-15-15.2-c17-0-26" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.37863493884913046 0 1.84123332281138158984622005720 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/20"] "2-15-15.2-c17-0-27" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.41258401492451274 0 2.14750379069910632445596253633 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/10"] "2-15-15.2-c17-0-28" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.4809341123718006 0 2.67357791847426170122902062768 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/31"] "2-15-15.2-c17-0-29" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.4530072181298835 0 2.73124435797928870852145851232 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/29"] "2-15-15.2-c17-0-3" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.19146850005148597 0 0.26098029302935565674380282764 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/7"] "2-15-15.2-c17-0-30" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.3253089145086952 0 2.79354969431056420975346244096 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/32"] "2-15-15.2-c17-0-31" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.1693025105932733 0 3.51420935623482184802613366523 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/28"] "2-15-15.2-c17-0-4" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.36178212283798805 0 0.33370118404400041422926006632 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/8"] "2-15-15.2-c17-0-5" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.2372746556035249 0 0.34050556701342350248236092107 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/1"] "2-15-15.2-c17-0-6" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.3537838025066716 0 0.36816907787618973418945526407 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/6"] "2-15-15.2-c17-0-7" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.16179568627481677 0 0.39789448008119846234189924464 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/19"] "2-15-15.2-c17-0-8" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false 0.4782639660112705 0 0.44486860438737865131256888967 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/21"] "2-15-15.2-c17-0-9" 5.242452966091729 27.48331310168397 2 15 "15.2" [] [[8.5, 0.0]] 17 true true false false -0.4660445999392145 0 0.47261869533465866398300697704 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/2/16"] "2-15-15.8-c17-0-0" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.1693025105932733 0 0.079403341014375122326511358702 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/28"] "2-15-15.8-c17-0-1" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.21325539906305174 0 0.12668878531039232680969466289 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/11"] "2-15-15.8-c17-0-10" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.3537838025066716 0 0.67291039125595601647382409850 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/6"] "2-15-15.8-c17-0-11" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.2986450423015509 0 0.69807414680934041370735908585 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/27"] "2-15-15.8-c17-0-12" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.3133382581878896 0 0.70274046805845888226964894656 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/4"] "2-15-15.8-c17-0-13" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.01456656305447067 0 0.73766470948419969013099943738 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/15"] "2-15-15.8-c17-0-14" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.36178212283798805 0 0.75469274643813902209171462409 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/8"] "2-15-15.8-c17-0-15" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.4530072181298835 0 0.899570113955247340341650160037 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/29"] "2-15-15.8-c17-0-16" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.16179568627481677 0 0.978903989283452690573516606770 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/19"] "2-15-15.8-c17-0-17" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.22559940967143932 0 0.993310537640875674011303046642 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/5"] "2-15-15.8-c17-0-18" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.4809341123718006 0 1.03490525831572428117267925030 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/31"] "2-15-15.8-c17-0-19" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.24613779900317928 0 1.19181884733000310449183881989 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/24"] "2-15-15.8-c17-0-2" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.41258401492451274 0 0.14388037491573258530021360532 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/10"] "2-15-15.8-c17-0-20" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.11811445718746881 0 1.37591473827386205799351195013 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/26"] "2-15-15.8-c17-0-21" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.3256307089094556 0 1.59021775677433450162196445072 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/9"] "2-15-15.8-c17-0-22" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.2372746556035249 0 1.61986585268014128229151944278 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/1"] "2-15-15.8-c17-0-23" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.12255244519713235 0 1.77413520542611925644893115468 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/25"] "2-15-15.8-c17-0-24" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.05798834814654406 0 1.79306268542200749055919645461 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/30"] "2-15-15.8-c17-0-25" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.14735168033145762 0 1.86329875349234549054685226822 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/23"] "2-15-15.8-c17-0-26" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.4874617875470529 0 1.99159920279618666754960425801 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/17"] "2-15-15.8-c17-0-27" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.4660445999392145 0 2.04974541903709867179694941892 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/16"] "2-15-15.8-c17-0-28" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.20639509639586523 0 2.07385729020894958042566832433 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/18"] "2-15-15.8-c17-0-29" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true 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5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.16940467305967666 0 0.26963292986092712684523680029 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/2"] "2-15-15.8-c17-0-6" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.126783254034373 0 0.56900461508885093774380005485 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/3"] "2-15-15.8-c17-0-7" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.08835387578163467 0 0.58349698420883429564423360992 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/13"] "2-15-15.8-c17-0-8" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false -0.37863493884913046 0 0.60430099529815031271425844847 ["ModularForm/GL2/Q/holomorphic/15/18/e/a/8/20"] "2-15-15.8-c17-0-9" 5.242452966091729 27.48331310168397 2 15 "15.8" [] [[8.5, 0.0]] 17 true true false false 0.039416581977112564 0 0.66662429369246716851354840955 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"5.4" [] [[8.5, 0.0]] 17 true true false false -0.17457584912648644 0 0.35588977836003927699248194563 ["ModularForm/GL2/Q/holomorphic/15/18/b/a/4/3"] "2-15-5.4-c17-0-4" 5.242452966091729 27.48331310168397 2 15 "5.4" [] [[8.5, 0.0]] 17 true true false false -0.24522500834909192 0 0.68865617857734179593006986994 ["ModularForm/GL2/Q/holomorphic/15/18/b/a/4/15"] "2-15-5.4-c17-0-5" 5.242452966091729 27.48331310168397 2 15 "5.4" [] [[8.5, 0.0]] 17 true true false false 0.24522500834909192 0 0.70259804881186751552768615231 ["ModularForm/GL2/Q/holomorphic/15/18/b/a/4/2"] "2-15-5.4-c17-0-6" 5.242452966091729 27.48331310168397 2 15 "5.4" [] [[8.5, 0.0]] 17 true true false false -0.4079868463873897 0 0.71993302567895903687067620302 ["ModularForm/GL2/Q/holomorphic/15/18/b/a/4/12"] "2-15-5.4-c17-0-7" 5.242452966091729 27.48331310168397 2 15 "5.4" [] [[8.5, 0.0]] 17 true true false false -0.05599719837171424 0 0.984442016370783391475045903277 ["ModularForm/GL2/Q/holomorphic/15/18/b/a/4/11"] "2-15-5.4-c17-0-8" 5.242452966091729 27.48331310168397 2 15 "5.4" [] [[8.5, 0.0]] 17 true true false false -0.245336613791394 0 1.10867777679357504826865616390 ["ModularForm/GL2/Q/holomorphic/15/18/b/a/4/9"] "2-15-5.4-c17-0-9" 5.242452966091729 27.48331310168397 2 15 "5.4" [] [[8.5, 0.0]] 17 true true false false 0.05599719837171424 0 1.16129961944234616298665031708 ["ModularForm/GL2/Q/holomorphic/15/18/b/a/4/6"] # 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).