# lfunc_search downloaded from the LMFDB on 24 August 2026. # Search link: https://www.lmfdb.org/L/2/2816/88.43/c1-0 # Query "{'degree': 2, 'conductor': 2816, 'spectral_label': 'c1-0'}" returned 344 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. "2-2816-1.1-c1-0-0" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.13756857783881879994422124457 ["ModularForm/GL2/Q/holomorphic/2816/2/a/t/1/3"] "2-2816-1.1-c1-0-1" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.20294255199200134888470598112 ["ModularForm/GL2/Q/holomorphic/2816/2/a/u/1/1"] "2-2816-1.1-c1-0-10" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.61032283955481647443119608852 ["ModularForm/GL2/Q/holomorphic/2816/2/a/d/1/1"] "2-2816-1.1-c1-0-11" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.61817793320342029024080208036 ["ModularForm/GL2/Q/holomorphic/2816/2/a/a/1/1"] "2-2816-1.1-c1-0-12" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.62381539672224599165275394802 ["ModularForm/GL2/Q/holomorphic/2816/2/a/h/1/1"] "2-2816-1.1-c1-0-13" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.64940160776818817791741155123 ["ModularForm/GL2/Q/holomorphic/2816/2/a/u/1/3"] "2-2816-1.1-c1-0-14" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.64948691180634922231554282190 ["ModularForm/GL2/Q/holomorphic/2816/2/a/v/1/3"] "2-2816-1.1-c1-0-15" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.65334955506962365015190325409 ["ModularForm/GL2/Q/holomorphic/2816/2/a/q/1/3"] "2-2816-1.1-c1-0-16" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.72424784418372901921636944716 ["ModularForm/GL2/Q/holomorphic/2816/2/a/a/1/2"] "2-2816-1.1-c1-0-17" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.73452814668345663719421791214 ["ModularForm/GL2/Q/holomorphic/2816/2/a/l/1/1"] "2-2816-1.1-c1-0-18" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.73720008377896155758676170668 ["ModularForm/GL2/Q/holomorphic/2816/2/a/t/1/1"] "2-2816-1.1-c1-0-19" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.76294282386679652378221676559 ["ModularForm/GL2/Q/holomorphic/2816/2/a/n/1/3"] "2-2816-1.1-c1-0-2" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.29657341059422946145935863651 ["ModularForm/GL2/Q/holomorphic/2816/2/a/q/1/2"] "2-2816-1.1-c1-0-20" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.77080078836883927768638573629 ["ModularForm/GL2/Q/holomorphic/2816/2/a/k/1/2"] "2-2816-1.1-c1-0-21" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.796363850156299018495010688099 ["ModularForm/GL2/Q/holomorphic/2816/2/a/u/1/4"] "2-2816-1.1-c1-0-22" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.883373063598070615849240824619 ["ModularForm/GL2/Q/holomorphic/2816/2/a/v/1/2"] "2-2816-1.1-c1-0-23" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.883921837265714319608854865205 ["ModularForm/GL2/Q/holomorphic/2816/2/a/s/1/1"] "2-2816-1.1-c1-0-24" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.901111208463514959700926931152 ["ModularForm/GL2/Q/holomorphic/2816/2/a/r/1/2"] "2-2816-1.1-c1-0-25" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.918045308686207430579323177776 ["ModularForm/GL2/Q/holomorphic/2816/2/a/o/1/1"] "2-2816-1.1-c1-0-26" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.943805578693414347452603734806 ["ModularForm/GL2/Q/holomorphic/2816/2/a/c/1/1"] "2-2816-1.1-c1-0-27" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.951647137693945711114743332926 ["ModularForm/GL2/Q/holomorphic/2816/2/a/u/1/2"] "2-2816-1.1-c1-0-28" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.952723904991995624602103848975 ["ModularForm/GL2/Q/holomorphic/2816/2/a/d/1/2"] "2-2816-1.1-c1-0-29" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.957951615779770089067848569129 ["ModularForm/GL2/Q/holomorphic/2816/2/a/q/1/1"] "2-2816-1.1-c1-0-3" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.31047411550073069793244394860 ["ModularForm/GL2/Q/holomorphic/2816/2/a/v/1/1"] "2-2816-1.1-c1-0-30" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.979605475550013581232755572785 ["ModularForm/GL2/Q/holomorphic/2816/2/a/q/1/4"] "2-2816-1.1-c1-0-31" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.989928506464978586324354401541 ["ModularForm/GL2/Q/holomorphic/2816/2/a/n/1/2"] "2-2816-1.1-c1-0-32" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.01558113287207679325689314659 ["ModularForm/GL2/Q/holomorphic/2816/2/a/f/1/1"] "2-2816-1.1-c1-0-33" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.01935450534989568951601772985 ["ModularForm/GL2/Q/holomorphic/2816/2/a/u/1/5"] "2-2816-1.1-c1-0-34" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.03794424854493168943066864709 ["ModularForm/GL2/Q/holomorphic/2816/2/a/t/1/5"] "2-2816-1.1-c1-0-35" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.05725595477103069968076693135 ["ModularForm/GL2/Q/holomorphic/2816/2/a/m/1/1"] "2-2816-1.1-c1-0-36" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.07725916679924796317980160369 ["ModularForm/GL2/Q/holomorphic/2816/2/a/v/1/4"] "2-2816-1.1-c1-0-37" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11538479112328901995386820226 ["ModularForm/GL2/Q/holomorphic/2816/2/a/r/1/4"] "2-2816-1.1-c1-0-38" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.12751396544681354092263299825 ["ModularForm/GL2/Q/holomorphic/2816/2/a/f/1/2"] "2-2816-1.1-c1-0-39" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.16292223478999860591772686857 ["ModularForm/GL2/Q/holomorphic/2816/2/a/p/1/2"] "2-2816-1.1-c1-0-4" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.38709579203208737082935588365 ["ModularForm/GL2/Q/holomorphic/2816/2/a/k/1/1"] "2-2816-1.1-c1-0-40" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.21819211801439956751414943066 ["ModularForm/GL2/Q/holomorphic/2816/2/a/t/1/4"] "2-2816-1.1-c1-0-41" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.22395868737334424138582808735 ["ModularForm/GL2/Q/holomorphic/2816/2/a/v/1/6"] "2-2816-1.1-c1-0-42" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.23643612758722216900275042894 ["ModularForm/GL2/Q/holomorphic/2816/2/a/p/1/1"] "2-2816-1.1-c1-0-43" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.24894739478090500449507239949 ["ModularForm/GL2/Q/holomorphic/2816/2/a/s/1/3"] "2-2816-1.1-c1-0-44" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.27199578960310589564259472466 ["ModularForm/GL2/Q/holomorphic/2816/2/a/j/1/1"] "2-2816-1.1-c1-0-45" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.28152079479250797638987386010 ["ModularForm/GL2/Q/holomorphic/2816/2/a/e/1/1"] "2-2816-1.1-c1-0-46" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.28307797385214965483565670317 ["ModularForm/GL2/Q/holomorphic/2816/2/a/b/1/1"] "2-2816-1.1-c1-0-47" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.28872686466183645163302138725 ["ModularForm/GL2/Q/holomorphic/2816/2/a/h/1/2"] "2-2816-1.1-c1-0-48" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.31331552586374309776712347157 ["ModularForm/GL2/Q/holomorphic/2816/2/a/t/1/6"] "2-2816-1.1-c1-0-49" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.34103855709644776449391753043 ["ModularForm/GL2/Q/holomorphic/2816/2/a/i/1/1"] "2-2816-1.1-c1-0-5" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.41377550735793894382686408593 ["ModularForm/GL2/Q/holomorphic/2816/2/a/n/1/1"] "2-2816-1.1-c1-0-50" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.35604488709546517135337957625 ["ModularForm/GL2/Q/holomorphic/2816/2/a/v/1/5"] "2-2816-1.1-c1-0-51" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.35991476838693617379470910063 ["ModularForm/GL2/Q/holomorphic/2816/2/a/k/1/3"] "2-2816-1.1-c1-0-52" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.41918972582913427335031483553 ["ModularForm/GL2/Q/holomorphic/2816/2/a/r/1/5"] "2-2816-1.1-c1-0-53" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.43879035861345555531278077750 ["ModularForm/GL2/Q/holomorphic/2816/2/a/m/1/2"] "2-2816-1.1-c1-0-54" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.45868451196800618376349640531 ["ModularForm/GL2/Q/holomorphic/2816/2/a/o/1/2"] "2-2816-1.1-c1-0-55" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.46547375576136080189230385737 ["ModularForm/GL2/Q/holomorphic/2816/2/a/l/1/3"] "2-2816-1.1-c1-0-56" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.46709256120846463975672314110 ["ModularForm/GL2/Q/holomorphic/2816/2/a/q/1/5"] "2-2816-1.1-c1-0-57" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.50528443120717451299076346950 ["ModularForm/GL2/Q/holomorphic/2816/2/a/n/1/4"] "2-2816-1.1-c1-0-58" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.50681757864329254079453344180 ["ModularForm/GL2/Q/holomorphic/2816/2/a/b/1/2"] "2-2816-1.1-c1-0-59" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.54303743875800522616303793963 ["ModularForm/GL2/Q/holomorphic/2816/2/a/u/1/6"] "2-2816-1.1-c1-0-6" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.47712851468440482021287824114 ["ModularForm/GL2/Q/holomorphic/2816/2/a/l/1/2"] "2-2816-1.1-c1-0-60" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.55588337750211028871837717046 ["ModularForm/GL2/Q/holomorphic/2816/2/a/m/1/4"] "2-2816-1.1-c1-0-61" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.55759137238645007758453324118 ["ModularForm/GL2/Q/holomorphic/2816/2/a/s/1/2"] "2-2816-1.1-c1-0-62" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.56132953902089484532396474512 ["ModularForm/GL2/Q/holomorphic/2816/2/a/g/1/1"] "2-2816-1.1-c1-0-63" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.59543730780179668373270484074 ["ModularForm/GL2/Q/holomorphic/2816/2/a/o/1/3"] "2-2816-1.1-c1-0-64" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.61053350867865588222701032982 ["ModularForm/GL2/Q/holomorphic/2816/2/a/s/1/4"] "2-2816-1.1-c1-0-65" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.65520783703195183552438940472 ["ModularForm/GL2/Q/holomorphic/2816/2/a/i/1/3"] "2-2816-1.1-c1-0-66" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.65728232978677999134144741557 ["ModularForm/GL2/Q/holomorphic/2816/2/a/j/1/2"] "2-2816-1.1-c1-0-67" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.66609184405031839178991573260 ["ModularForm/GL2/Q/holomorphic/2816/2/a/m/1/3"] "2-2816-1.1-c1-0-68" 4.741927161945085 22.48587320919257 2 2816 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 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["ModularForm/GL2/Q/holomorphic/2816/2/g/f/1407/11"] "2-2816-88.43-c1-0-85" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false -0.4352244953876788 0 1.69389950529068788701858116716 ["ModularForm/GL2/Q/holomorphic/2816/2/g/f/1407/3"] "2-2816-88.43-c1-0-86" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false 0.27327649802748394 0 1.72901748908692849479324932322 ["ModularForm/GL2/Q/holomorphic/2816/2/g/h/1407/11"] "2-2816-88.43-c1-0-87" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false 0.45875807706845334 0 1.75103465626412802611631650815 ["ModularForm/GL2/Q/holomorphic/2816/2/g/g/1407/7"] "2-2816-88.43-c1-0-88" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false 0.38202592557542436 0 1.84258951826827993163152621501 ["ModularForm/GL2/Q/holomorphic/2816/2/g/d/1407/10"] "2-2816-88.43-c1-0-89" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false -0.48513376293527966 0 1.91753207426450369096957999906 ["ModularForm/GL2/Q/holomorphic/2816/2/g/e/1407/9"] "2-2816-88.43-c1-0-9" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false -0.2206454286092323 0 0.23462742095237348917116169591 ["ModularForm/GL2/Q/holomorphic/2816/2/g/f/1407/7"] "2-2816-88.43-c1-0-90" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false 0.375 0 1.96430246124001863987949287617 ["ModularForm/GL2/Q/holomorphic/2816/2/g/c/1407/7"] "2-2816-88.43-c1-0-91" 4.741927161945085 22.48587320919257 2 2816 "88.43" [] [[0.5, 0.0]] 1 true true false false -0.4352244953876788 0 2.17719373657928333943264004825 ["ModularForm/GL2/Q/holomorphic/2816/2/g/g/1407/9"] # 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).