# lfunc_search downloaded from the LMFDB on 25 May 2026. # Search link: https://www.lmfdb.org/L/2/675/5.4/c3-0 # Query "{'degree': 2, 'conductor': 675, 'spectral_label': 'c3-0'}" returned 148 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. "2-675-1.1-c3-0-0" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.29994959236767997394842154716 ["ModularForm/GL2/Q/holomorphic/675/4/a/v/1/1"] "2-675-1.1-c3-0-1" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.0 0 0.30485927730869586333584222951 ["ModularForm/GL2/Q/holomorphic/675/4/a/d/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/d"] "2-675-1.1-c3-0-10" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.53723113977590312906993527336 ["ModularForm/GL2/Q/holomorphic/675/4/a/y/1/1"] "2-675-1.1-c3-0-11" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.59800648615558948276356389523 ["ModularForm/GL2/Q/holomorphic/675/4/a/s/1/2"] "2-675-1.1-c3-0-12" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.64435053794068204780847327356 ["ModularForm/GL2/Q/holomorphic/675/4/a/v/1/3"] "2-675-1.1-c3-0-13" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.66646825403358158014157779794 ["ModularForm/GL2/Q/holomorphic/675/4/a/bc/1/3"] "2-675-1.1-c3-0-14" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.0 0 0.67967671919860592609924455428 ["ModularForm/GL2/Q/holomorphic/675/4/a/b/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/b"] "2-675-1.1-c3-0-15" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.68111586178742939065748965356 ["ModularForm/GL2/Q/holomorphic/675/4/a/y/1/2"] "2-675-1.1-c3-0-16" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.70296695503869248308554323077 ["ModularForm/GL2/Q/holomorphic/675/4/a/bc/1/4"] "2-675-1.1-c3-0-17" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.73835726156009355318343411011 ["ModularForm/GL2/Q/holomorphic/675/4/a/r/1/1"] "2-675-1.1-c3-0-18" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.74066201650976485270336472157 ["ModularForm/GL2/Q/holomorphic/675/4/a/y/1/3"] "2-675-1.1-c3-0-19" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.74285780043883536650066637100 ["ModularForm/GL2/Q/holomorphic/675/4/a/w/1/3"] "2-675-1.1-c3-0-2" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.31926770527107972592090960562 ["ModularForm/GL2/Q/holomorphic/675/4/a/w/1/2"] "2-675-1.1-c3-0-20" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.810388981678696396101618511468 ["ModularForm/GL2/Q/holomorphic/675/4/a/w/1/1"] "2-675-1.1-c3-0-21" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.826904054479456536882631703960 ["ModularForm/GL2/Q/holomorphic/675/4/a/s/1/1"] "2-675-1.1-c3-0-22" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.852461475986206196179570798605 ["ModularForm/GL2/Q/holomorphic/675/4/a/x/1/3"] "2-675-1.1-c3-0-23" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.888135214714639769298303840061 ["ModularForm/GL2/Q/holomorphic/675/4/a/t/1/4"] "2-675-1.1-c3-0-24" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.0 0 0.905007968081566446347284821587 ["ModularForm/GL2/Q/holomorphic/675/4/a/g/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/g"] "2-675-1.1-c3-0-25" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.936627922637311638810939463396 ["ModularForm/GL2/Q/holomorphic/675/4/a/v/1/2"] "2-675-1.1-c3-0-26" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.937882546359165829616370828242 ["ModularForm/GL2/Q/holomorphic/675/4/a/t/1/2"] "2-675-1.1-c3-0-27" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.970419999617048936849690887496 ["ModularForm/GL2/Q/holomorphic/675/4/a/o/1/1"] "2-675-1.1-c3-0-28" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.970705560094785034991030399450 ["ModularForm/GL2/Q/holomorphic/675/4/a/x/1/2"] "2-675-1.1-c3-0-29" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.973003085835479335974565493895 ["ModularForm/GL2/Q/holomorphic/675/4/a/bc/1/5"] "2-675-1.1-c3-0-3" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.39703553741338774913318063482 ["ModularForm/GL2/Q/holomorphic/675/4/a/t/1/3"] "2-675-1.1-c3-0-30" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.973513462697131407781124100073 ["ModularForm/GL2/Q/holomorphic/675/4/a/p/1/3"] "2-675-1.1-c3-0-31" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.980669203994164410294914505573 ["ModularForm/GL2/Q/holomorphic/675/4/a/bc/1/1"] "2-675-1.1-c3-0-32" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.0 0 1.03211900351115389901864813923 ["ModularForm/GL2/Q/holomorphic/675/4/a/j/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/j"] "2-675-1.1-c3-0-33" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.09222483462991891044130810889 ["ModularForm/GL2/Q/holomorphic/675/4/a/bb/1/1"] "2-675-1.1-c3-0-34" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.13238179255917717196505540243 ["ModularForm/GL2/Q/holomorphic/675/4/a/q/1/1"] "2-675-1.1-c3-0-35" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.14127977139088805928683357548 ["ModularForm/GL2/Q/holomorphic/675/4/a/u/1/2"] "2-675-1.1-c3-0-36" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.21665495723127506118951012085 ["ModularForm/GL2/Q/holomorphic/675/4/a/n/1/1"] "2-675-1.1-c3-0-37" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.21980592044597267614797041905 ["ModularForm/GL2/Q/holomorphic/675/4/a/l/1/1"] "2-675-1.1-c3-0-38" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.26028489473052721435051228660 ["ModularForm/GL2/Q/holomorphic/675/4/a/ba/1/1"] "2-675-1.1-c3-0-39" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.26508012409001347136225239196 ["ModularForm/GL2/Q/holomorphic/675/4/a/k/1/2"] "2-675-1.1-c3-0-4" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.43724613678547454417322215859 ["ModularForm/GL2/Q/holomorphic/675/4/a/r/1/2"] "2-675-1.1-c3-0-40" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.26709928904406110050944629086 ["ModularForm/GL2/Q/holomorphic/675/4/a/z/1/2"] "2-675-1.1-c3-0-41" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.26835289995426341540392963987 ["ModularForm/GL2/Q/holomorphic/675/4/a/k/1/1"] "2-675-1.1-c3-0-42" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.28118990164797353014875740776 ["ModularForm/GL2/Q/holomorphic/675/4/a/bb/1/3"] "2-675-1.1-c3-0-43" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.28298752287555848132849144909 ["ModularForm/GL2/Q/holomorphic/675/4/a/w/1/4"] "2-675-1.1-c3-0-44" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.28324109001328204163271261247 ["ModularForm/GL2/Q/holomorphic/675/4/a/x/1/4"] "2-675-1.1-c3-0-45" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.28831261844016775940692971992 ["ModularForm/GL2/Q/holomorphic/675/4/a/r/1/3"] "2-675-1.1-c3-0-46" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.31521111768489924789725266447 ["ModularForm/GL2/Q/holomorphic/675/4/a/m/1/1"] "2-675-1.1-c3-0-47" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.32324874297173100205937929991 ["ModularForm/GL2/Q/holomorphic/675/4/a/q/1/2"] "2-675-1.1-c3-0-48" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 1.32478632933604666724255491577 ["ModularForm/GL2/Q/holomorphic/675/4/a/c/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/c"] "2-675-1.1-c3-0-49" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 1.33231959963257179621788545111 ["ModularForm/GL2/Q/holomorphic/675/4/a/e/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/e"] "2-675-1.1-c3-0-5" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.44311902973844939756212537546 ["ModularForm/GL2/Q/holomorphic/675/4/a/t/1/1"] "2-675-1.1-c3-0-50" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.34709346063123769519462392363 ["ModularForm/GL2/Q/holomorphic/675/4/a/y/1/4"] "2-675-1.1-c3-0-51" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.35769056484624897502867917158 ["ModularForm/GL2/Q/holomorphic/675/4/a/v/1/4"] "2-675-1.1-c3-0-52" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.36847008875610006275517218067 ["ModularForm/GL2/Q/holomorphic/675/4/a/z/1/1"] "2-675-1.1-c3-0-53" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 1.38147807328020856922759122332 ["ModularForm/GL2/Q/holomorphic/675/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/a"] "2-675-1.1-c3-0-54" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.41017642309941025125754633671 ["ModularForm/GL2/Q/holomorphic/675/4/a/bb/1/2"] "2-675-1.1-c3-0-55" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.43917331269640040388040704134 ["ModularForm/GL2/Q/holomorphic/675/4/a/ba/1/2"] "2-675-1.1-c3-0-56" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 1.44265296518786514317146173828 ["ModularForm/GL2/Q/holomorphic/675/4/a/f/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/f"] "2-675-1.1-c3-0-57" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.48098997914736716194646824516 ["ModularForm/GL2/Q/holomorphic/675/4/a/o/1/2"] "2-675-1.1-c3-0-58" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.49099921031827504506508565968 ["ModularForm/GL2/Q/holomorphic/675/4/a/bc/1/6"] "2-675-1.1-c3-0-59" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.49467729914544331304617112951 ["ModularForm/GL2/Q/holomorphic/675/4/a/u/1/1"] "2-675-1.1-c3-0-6" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.44543315397134557564921354005 ["ModularForm/GL2/Q/holomorphic/675/4/a/bc/1/2"] "2-675-1.1-c3-0-60" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.50772884242069918677698866224 ["ModularForm/GL2/Q/holomorphic/675/4/a/bb/1/4"] "2-675-1.1-c3-0-61" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 1.52704583141679564805674854093 ["ModularForm/GL2/Q/holomorphic/675/4/a/h/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/h"] "2-675-1.1-c3-0-62" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.54044033692394974397925737092 ["ModularForm/GL2/Q/holomorphic/675/4/a/s/1/3"] "2-675-1.1-c3-0-63" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.57132536697978419394816103875 ["ModularForm/GL2/Q/holomorphic/675/4/a/u/1/3"] "2-675-1.1-c3-0-64" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 1.63729010067989467341178877996 ["ModularForm/GL2/Q/holomorphic/675/4/a/i/1/1", "ModularForm/GL2/Q/holomorphic/675/4/a/i"] "2-675-1.1-c3-0-65" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.82281898029676283071440272758 ["ModularForm/GL2/Q/holomorphic/675/4/a/z/1/3"] "2-675-1.1-c3-0-66" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.95046382304225445587043269107 ["ModularForm/GL2/Q/holomorphic/675/4/a/l/1/2"] "2-675-1.1-c3-0-67" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.96027919670111924256791156883 ["ModularForm/GL2/Q/holomorphic/675/4/a/ba/1/3"] "2-675-1.1-c3-0-68" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 1.98276231883436941238507234363 ["ModularForm/GL2/Q/holomorphic/675/4/a/ba/1/4"] "2-675-1.1-c3-0-69" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 2.02674962359235680984445124316 ["ModularForm/GL2/Q/holomorphic/675/4/a/m/1/2"] "2-675-1.1-c3-0-7" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.46378328550145000007886878545 ["ModularForm/GL2/Q/holomorphic/675/4/a/x/1/1"] "2-675-1.1-c3-0-70" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 2.08984126771056146751994376033 ["ModularForm/GL2/Q/holomorphic/675/4/a/bb/1/5"] "2-675-1.1-c3-0-71" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 2.14251290417858218891827752121 ["ModularForm/GL2/Q/holomorphic/675/4/a/n/1/2"] "2-675-1.1-c3-0-72" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 2.15127974809302038785940115539 ["ModularForm/GL2/Q/holomorphic/675/4/a/z/1/4"] "2-675-1.1-c3-0-73" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 2.28720332740287806150101776029 ["ModularForm/GL2/Q/holomorphic/675/4/a/q/1/3"] "2-675-1.1-c3-0-74" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 2.33989125781742126681501719437 ["ModularForm/GL2/Q/holomorphic/675/4/a/bb/1/6"] "2-675-1.1-c3-0-75" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.5 1 2.35469461794545330899915873289 ["ModularForm/GL2/Q/holomorphic/675/4/a/u/1/4"] "2-675-1.1-c3-0-8" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.48757158948022444979284534528 ["ModularForm/GL2/Q/holomorphic/675/4/a/p/1/2"] "2-675-1.1-c3-0-9" 6.310807337724305 39.82628925387493 2 675 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.51731656117783109209655973398 ["ModularForm/GL2/Q/holomorphic/675/4/a/p/1/1"] "2-675-5.4-c3-0-0" 6.310807337724305 39.82628925387493 2 675 "5.4" [] [[1.5, 0.0]] 3 true true false false 0.42620819117478337 0 0.04050680214093530691639218049 ["ModularForm/GL2/Q/holomorphic/675/4/b/m/649/4"] "2-675-5.4-c3-0-1" 6.310807337724305 39.82628925387493 2 675 "5.4" [] [[1.5, 0.0]] 3 true true false false 0.17620819117478337 0 0.13502552357486427154708811094 ["ModularForm/GL2/Q/holomorphic/675/4/b/o/649/7"] "2-675-5.4-c3-0-10" 6.310807337724305 39.82628925387493 2 675 "5.4" [] [[1.5, 0.0]] 3 true true false false 0.42620819117478337 0 0.29955255620455492871205772705 ["ModularForm/GL2/Q/holomorphic/675/4/b/l/649/6"] "2-675-5.4-c3-0-11" 6.310807337724305 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false false -0.07379180882521663 0 2.09991871873655084352031644837 ["ModularForm/GL2/Q/holomorphic/675/4/b/k/649/1"] "2-675-5.4-c3-0-71" 6.310807337724305 39.82628925387493 2 675 "5.4" [] [[1.5, 0.0]] 3 true true false false -0.32379180882521663 0 2.14477517144774131398854568448 ["ModularForm/GL2/Q/holomorphic/675/4/b/p/649/3"] "2-675-5.4-c3-0-8" 6.310807337724305 39.82628925387493 2 675 "5.4" [] [[1.5, 0.0]] 3 true true false false 0.17620819117478337 0 0.24048957974099074408105439362 ["ModularForm/GL2/Q/holomorphic/675/4/b/q/649/7"] "2-675-5.4-c3-0-9" 6.310807337724305 39.82628925387493 2 675 "5.4" [] [[1.5, 0.0]] 3 true true false false 0.32379180882521663 0 0.28394670929402285536872900740 ["ModularForm/GL2/Q/holomorphic/675/4/b/j/649/4"] # 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).