# lfunc_search downloaded from the LMFDB on 30 May 2026. # Search link: https://www.lmfdb.org/L/1/197/197.178/r0-0 # Query "{'degree': 1, 'conductor': 197, 'spectral_label': 'r0-0'}" returned 97 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. "1-197-197.10-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.10" [[0, 0.0]] [] 0 true true false false 0.12946221774030392 0 1.67184123068 ["Character/Dirichlet/197/10"] "1-197-197.100-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.100" [[0, 0.0]] [] 0 true true false false -0.09033592254641888 0 1.49202651632 ["Character/Dirichlet/197/100"] "1-197-197.101-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.101" [[0, 0.0]] [] 0 true true false false 0.26649814388577203 0 1.52275789613 ["Character/Dirichlet/197/101"] "1-197-197.104-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.104" [[0, 0.0]] [] 0 true true false false -0.04359602502980999 0 1.28730267287 ["Character/Dirichlet/197/104"] "1-197-197.105-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.105" [[0, 0.0]] [] 0 true true false false 0.24967010187873329 0 2.15877076128 ["Character/Dirichlet/197/105"] "1-197-197.107-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.107" [[0, 0.0]] [] 0 true true false false -0.4528162206353165 0 1.04822530339 ["Character/Dirichlet/197/107"] "1-197-197.109-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.109" [[0, 0.0]] [] 0 true true false false -0.07810039665872964 0 1.68690009992 ["Character/Dirichlet/197/109"] "1-197-197.112-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.112" [[0, 0.0]] [] 0 true true false false -0.21855385523191645 0 1.06715690888 ["Character/Dirichlet/197/112"] "1-197-197.114-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.114" [[0, 0.0]] [] 0 true true false false -0.32724887216393994 0 0.430615882632 ["Character/Dirichlet/197/114"] "1-197-197.116-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.116" [[0, 0.0]] [] 0 true true false false 0.4528162206353165 0 2.99889710643 ["Character/Dirichlet/197/116"] "1-197-197.121-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.121" [[0, 0.0]] [] 0 true true false false -0.30859478992233974 0 2.2277426902 ["Character/Dirichlet/197/121"] "1-197-197.127-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.127" [[0, 0.0]] [] 0 true true false false 0.30859478992233974 0 0.0739869204955 ["Character/Dirichlet/197/127"] "1-197-197.132-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.132" [[0, 0.0]] [] 0 true true false false 0.09033592254641888 0 1.91559434138 ["Character/Dirichlet/197/132"] "1-197-197.133-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.133" [[0, 0.0]] [] 0 true true false false 0.2739288761903046 0 2.21401387096 ["Character/Dirichlet/197/133"] "1-197-197.134-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.134" [[0, 0.0]] [] 0 true true false false 0.30930385337461413 0 2.24928482514 ["Character/Dirichlet/197/134"] "1-197-197.135-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.135" [[0, 0.0]] [] 0 true true false false -0.14741646155713956 0 0.64551711121 ["Character/Dirichlet/197/135"] "1-197-197.136-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.136" [[0, 0.0]] [] 0 true true false false 0.29408803404577666 0 0.293750542631 ["Character/Dirichlet/197/136"] "1-197-197.137-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.137" [[0, 0.0]] [] 0 true true false false -0.403045409687069 0 1.04238786232 ["Character/Dirichlet/197/137"] "1-197-197.138-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.138" [[0, 0.0]] [] 0 true true false false -0.12946221774030392 0 1.33435878099 ["Character/Dirichlet/197/138"] "1-197-197.142-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.142" [[0, 0.0]] [] 0 true true false false 0.2276633254603835 0 1.80425434217 ["Character/Dirichlet/197/142"] "1-197-197.143-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.143" [[0, 0.0]] [] 0 true true false false 0.0350649863938231 0 1.08318736342 ["Character/Dirichlet/197/143"] "1-197-197.144-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.144" [[0, 0.0]] [] 0 true true false false -0.1847958340550544 0 1.58321719098 ["Character/Dirichlet/197/144"] "1-197-197.146-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.146" [[0, 0.0]] [] 0 true true false false 0.21855385523191645 0 1.22540697869 ["Character/Dirichlet/197/146"] "1-197-197.148-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.148" [[0, 0.0]] [] 0 true true false false 0.11402767656383739 0 2.3489614966 ["Character/Dirichlet/197/148"] "1-197-197.15-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.15" [[0, 0.0]] [] 0 true true false false 0.3594974266558199 0 1.38236289856 ["Character/Dirichlet/197/15"] "1-197-197.150-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.150" [[0, 0.0]] [] 0 true true false false -0.037174433415758414 0 1.2592450121 ["Character/Dirichlet/197/150"] "1-197-197.154-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.154" [[0, 0.0]] [] 0 true true false false -0.2276633254603835 0 1.35431045441 ["Character/Dirichlet/197/154"] "1-197-197.155-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.155" [[0, 0.0]] [] 0 true true false false -0.29408803404577666 0 3.01809624252 ["Character/Dirichlet/197/155"] "1-197-197.156-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.156" [[0, 0.0]] [] 0 true true false false 0.1895389993844341 0 2.28202727147 ["Character/Dirichlet/197/156"] "1-197-197.157-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.157" [[0, 0.0]] [] 0 true true false false 0.14561436660178886 0 1.81230677886 ["Character/Dirichlet/197/157"] "1-197-197.158-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.158" [[0, 0.0]] [] 0 true true false false -0.26649814388577203 0 0.877637231817 ["Character/Dirichlet/197/158"] "1-197-197.16-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.16" [[0, 0.0]] [] 0 true true false false -0.16632072869304154 0 1.44058698327 ["Character/Dirichlet/197/16"] "1-197-197.160-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.160" [[0, 0.0]] [] 0 true true false false -0.26366988938792757 0 1.02411950535 ["Character/Dirichlet/197/160"] "1-197-197.161-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.161" [[0, 0.0]] [] 0 true true false false -0.08513061147946419 0 0.925980201103 ["Character/Dirichlet/197/161"] "1-197-197.163-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.163" [[0, 0.0]] [] 0 true true false false 0.4799229936663894 0 2.10067682948 ["Character/Dirichlet/197/163"] "1-197-197.164-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.164" [[0, 0.0]] [] 0 true true false false -0.34517569036422074 0 0.377104676573 ["Character/Dirichlet/197/164"] "1-197-197.168-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.168" [[0, 0.0]] [] 0 true true false false -0.4799229936663894 0 0.982914972912 ["Character/Dirichlet/197/168"] "1-197-197.169-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.169" [[0, 0.0]] [] 0 true true false false -0.4478346626205766 0 2.38203098988 ["Character/Dirichlet/197/169"] "1-197-197.171-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.171" [[0, 0.0]] [] 0 true true false false -0.4882314699797721 0 0.000792115001628 ["Character/Dirichlet/197/171"] "1-197-197.172-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.172" [[0, 0.0]] [] 0 true true false false -0.10783020210828213 0 0.883385701832 ["Character/Dirichlet/197/172"] "1-197-197.173-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.173" [[0, 0.0]] [] 0 true true false false 0.06231787709753754 0 1.06931665116 ["Character/Dirichlet/197/173"] "1-197-197.174-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.174" [[0, 0.0]] [] 0 true true false false 0.403045409687069 0 2.51472162408 ["Character/Dirichlet/197/174"] "1-197-197.175-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.175" [[0, 0.0]] [] 0 true true false false 0.2327539075374993 0 2.10216831576 ["Character/Dirichlet/197/175"] "1-197-197.178-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.178" [[0, 0.0]] [] 0 true true false false 0.32724887216393994 0 1.41921174856 ["Character/Dirichlet/197/178"] "1-197-197.181-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.181" [[0, 0.0]] [] 0 true true false false 0.26366988938792757 0 1.77764049814 ["Character/Dirichlet/197/181"] "1-197-197.182-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.182" [[0, 0.0]] [] 0 true true false false -0.24967010187873329 0 1.68235877367 ["Character/Dirichlet/197/182"] "1-197-197.187-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.187" [[0, 0.0]] [] 0 true true false false -0.1381654872254983 0 1.3287530446 ["Character/Dirichlet/197/187"] "1-197-197.188-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.188" [[0, 0.0]] [] 0 true true false false -0.2327539075374993 0 0.816580508922 ["Character/Dirichlet/197/188"] "1-197-197.19-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.19" [[0, 0.0]] [] 0 true true false false -0.24385313395673214 0 1.04318566428 ["Character/Dirichlet/197/19"] "1-197-197.190-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.190" [[0, 0.0]] [] 0 true true false false 0.13507063440658681 0 0.976006583962 ["Character/Dirichlet/197/190"] "1-197-197.191-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.191" [[0, 0.0]] [] 0 true true false false 0.34517569036422074 0 1.51538086326 ["Character/Dirichlet/197/191"] "1-197-197.193-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.193" [[0, 0.0]] [] 0 true true false false -0.07270121539451029 0 0.502499354596 ["Character/Dirichlet/197/193"] "1-197-197.196-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.196" [[0, 0.0]] [] 0 true true true true 0.0 0 0.826247117918 ["Character/Dirichlet/197/196"] "1-197-197.22-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.22" [[0, 0.0]] [] 0 true true false false -0.47325670100353423 0 2.33917923407 ["Character/Dirichlet/197/22"] "1-197-197.23-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.23" [[0, 0.0]] [] 0 true true false false 0.22504833283742415 0 1.69723413645 ["Character/Dirichlet/197/23"] "1-197-197.24-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.24" [[0, 0.0]] [] 0 true true false false -0.1895389993844341 0 1.62520820451 ["Character/Dirichlet/197/24"] "1-197-197.25-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.25" [[0, 0.0]] [] 0 true true false false -0.30930385337461413 0 0.800951411427 ["Character/Dirichlet/197/25"] "1-197-197.26-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.26" [[0, 0.0]] [] 0 true true false false 0.1847958340550544 0 2.05209257637 ["Character/Dirichlet/197/26"] "1-197-197.28-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.28" [[0, 0.0]] [] 0 true true false false -0.13507063440658681 0 1.36973309958 ["Character/Dirichlet/197/28"] "1-197-197.29-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.29" [[0, 0.0]] [] 0 true true false false -0.3991924751293487 0 2.94662322175 ["Character/Dirichlet/197/29"] "1-197-197.33-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.33" [[0, 0.0]] [] 0 true true false false 0.15872252247726792 0 2.24954544193 ["Character/Dirichlet/197/33"] "1-197-197.34-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.34" [[0, 0.0]] [] 0 true true false false 0.3991924751293487 0 0.587129967934 ["Character/Dirichlet/197/34"] "1-197-197.36-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.36" [[0, 0.0]] [] 0 true true false false 0.04359602502980999 0 2.08329081291 ["Character/Dirichlet/197/36"] "1-197-197.37-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.37" [[0, 0.0]] [] 0 true true false false 0.16632072869304154 0 2.23105417812 ["Character/Dirichlet/197/37"] "1-197-197.39-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.39" [[0, 0.0]] [] 0 true true false false 0.39212065301962495 0 2.75757967856 ["Character/Dirichlet/197/39"] "1-197-197.4-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.4" [[0, 0.0]] [] 0 true true false false -0.11402767656383739 0 1.79466053495 ["Character/Dirichlet/197/4"] "1-197-197.40-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.40" [[0, 0.0]] [] 0 true true false false -0.2739288761903046 0 0.74646965901 ["Character/Dirichlet/197/40"] "1-197-197.41-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.41" [[0, 0.0]] [] 0 true true false false -0.06231787709753754 0 1.4182276789 ["Character/Dirichlet/197/41"] "1-197-197.42-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.42" [[0, 0.0]] [] 0 true true false false -0.2856046490019268 0 1.17438197434 ["Character/Dirichlet/197/42"] "1-197-197.43-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.43" [[0, 0.0]] [] 0 true true false false 0.256333215450189 0 1.72382368347 ["Character/Dirichlet/197/43"] "1-197-197.47-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.47" [[0, 0.0]] [] 0 true true false false 0.07810039665872964 0 1.52014550894 ["Character/Dirichlet/197/47"] "1-197-197.49-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.49" [[0, 0.0]] [] 0 true true false false 0.07270121539451029 0 1.05430897391 ["Character/Dirichlet/197/49"] "1-197-197.51-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.51" [[0, 0.0]] [] 0 true true false false 0.03812951347573815 0 1.91584777108 ["Character/Dirichlet/197/51"] "1-197-197.53-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.53" [[0, 0.0]] [] 0 true true false false 0.4882314699797721 0 1.36201957425 ["Character/Dirichlet/197/53"] "1-197-197.54-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.54" [[0, 0.0]] [] 0 true true false false 0.14741646155713956 0 1.85340558389 ["Character/Dirichlet/197/54"] "1-197-197.55-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.55" [[0, 0.0]] [] 0 true true false false -0.256333215450189 0 0.996436921887 ["Character/Dirichlet/197/55"] "1-197-197.59-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.59" [[0, 0.0]] [] 0 true true false false 0.1381654872254983 0 1.6169304834 ["Character/Dirichlet/197/59"] "1-197-197.6-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.6" [[0, 0.0]] [] 0 true true false false -0.15872252247726792 0 1.44221904864 ["Character/Dirichlet/197/6"] "1-197-197.60-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.60" [[0, 0.0]] [] 0 true true false false -0.22504833283742415 0 0.159559355883 ["Character/Dirichlet/197/60"] "1-197-197.61-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.61" [[0, 0.0]] [] 0 true true false false 0.2856046490019268 0 1.74844284825 ["Character/Dirichlet/197/61"] "1-197-197.62-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.62" [[0, 0.0]] [] 0 true true false false -0.0350649863938231 0 1.38761417763 ["Character/Dirichlet/197/62"] "1-197-197.63-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.63" [[0, 0.0]] [] 0 true true false false 0.10783020210828213 0 1.51450577234 ["Character/Dirichlet/197/63"] "1-197-197.64-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.64" [[0, 0.0]] [] 0 true true false false -0.14561436660178886 0 0.993858966527 ["Character/Dirichlet/197/64"] "1-197-197.65-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.65" [[0, 0.0]] [] 0 true true false false -0.21642679525310946 0 0.709340550987 ["Character/Dirichlet/197/65"] "1-197-197.7-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.7" [[0, 0.0]] [] 0 true true false false 0.4478346626205766 0 0.525527290391 ["Character/Dirichlet/197/7"] "1-197-197.70-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.70" [[0, 0.0]] [] 0 true true false false 0.17541174266325574 0 1.75713798528 ["Character/Dirichlet/197/70"] "1-197-197.76-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.76" [[0, 0.0]] [] 0 true true false false -0.17541174266325574 0 1.64628211264 ["Character/Dirichlet/197/76"] "1-197-197.81-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.81" [[0, 0.0]] [] 0 true true false false -0.06561965551355697 0 1.28499548736 ["Character/Dirichlet/197/81"] "1-197-197.83-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.83" [[0, 0.0]] [] 0 true true false false 0.24385313395673214 0 1.87128021338 ["Character/Dirichlet/197/83"] "1-197-197.85-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.85" [[0, 0.0]] [] 0 true true false false -0.03812951347573815 0 0.8497072006 ["Character/Dirichlet/197/85"] "1-197-197.88-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.88" [[0, 0.0]] [] 0 true true false false 0.037174433415758414 0 0.876047309966 ["Character/Dirichlet/197/88"] "1-197-197.9-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.9" [[0, 0.0]] [] 0 true true false false 0.47325670100353423 0 0.0934450019583 ["Character/Dirichlet/197/9"] "1-197-197.90-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.90" [[0, 0.0]] [] 0 true true false false 0.06561965551355697 0 0.663131844056 ["Character/Dirichlet/197/90"] "1-197-197.92-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.92" [[0, 0.0]] [] 0 true true false false -0.3594974266558199 0 0.65865493026 ["Character/Dirichlet/197/92"] "1-197-197.93-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.93" [[0, 0.0]] [] 0 true true false false 0.08513061147946419 0 1.31227420881 ["Character/Dirichlet/197/93"] "1-197-197.96-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.96" [[0, 0.0]] [] 0 true true false false -0.39212065301962495 0 0.794346172922 ["Character/Dirichlet/197/96"] "1-197-197.97-r0-0-0" 0.9148641538305795 0.9148641538305795 1 197 "197.97" [[0, 0.0]] [] 0 true true false false 0.21642679525310946 0 1.70130805228 ["Character/Dirichlet/197/97"] # 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).