Properties

Degree 1
Conductor 89
Sign $0.660 - 0.750i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.841 − 0.540i)2-s + (0.909 − 0.415i)3-s + (0.415 − 0.909i)4-s + (0.142 + 0.989i)5-s + (0.540 − 0.841i)6-s + (−0.989 + 0.142i)7-s + (−0.142 − 0.989i)8-s + (0.654 − 0.755i)9-s + (0.654 + 0.755i)10-s + (−0.142 + 0.989i)11-s i·12-s + (−0.909 + 0.415i)13-s + (−0.755 + 0.654i)14-s + (0.540 + 0.841i)15-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)17-s + ⋯
L(s,χ)  = 1  + (0.841 − 0.540i)2-s + (0.909 − 0.415i)3-s + (0.415 − 0.909i)4-s + (0.142 + 0.989i)5-s + (0.540 − 0.841i)6-s + (−0.989 + 0.142i)7-s + (−0.142 − 0.989i)8-s + (0.654 − 0.755i)9-s + (0.654 + 0.755i)10-s + (−0.142 + 0.989i)11-s i·12-s + (−0.909 + 0.415i)13-s + (−0.755 + 0.654i)14-s + (0.540 + 0.841i)15-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.660 - 0.750i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.660 - 0.750i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.660 - 0.750i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (21, \cdot )$
Sato-Tate  :  $\mu(44)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 89,\ (0:\ ),\ 0.660 - 0.750i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.633007101 - 0.7382562632i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.633007101 - 0.7382562632i\)
\(L(\chi,1)\)  \(\approx\)  \(1.685289393 - 0.5697825519i\)
\(L(1,\chi)\)  \(\approx\)  \(1.685289393 - 0.5697825519i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−31.107000840400357174581466544414, −29.75812719585743694116714496555, −28.832369977495944472543776928667, −27.102976920463142479744946872334, −26.31248606014543580972442702755, −25.07585750639897588169466912035, −24.63866094704444012232236413789, −23.34479467016783914228369027481, −21.9801846060404336083106026651, −21.26265582820026060412208409368, −20.1321236823662080258950168998, −19.29038468870665925822917521792, −17.145952255963093126091226593819, −16.268365111222899396322197135084, −15.44356005857221297516101010296, −14.1720108559689836851560041264, −13.15189756742455887049026674375, −12.476143848727603518695585444962, −10.46884315368397638365300949669, −8.95330134744128491634141762911, −8.1142414923373455551394058076, −6.51334204731557476831486469508, −5.04321601169484943665681990121, −3.86239846100532881359260545404, −2.58550026582726703548981240784, 2.25917748017797984374966874233, 2.96083412827119223571645419181, 4.46503437095183538920807141189, 6.53241529812561843015462419478, 7.13439628149696372076140982136, 9.36950217311456810123341655819, 10.16918249206535294941175390578, 11.77114007490005612940103780076, 12.93915836465682557160010546979, 13.76268107671948964676107906301, 14.95302825195012228590214442695, 15.52774644829846216199192434779, 17.759853877318785966220887025366, 19.15480698340495358360057327740, 19.4845932819693584446985178615, 20.77405601933705688908335882384, 21.952849900626556127718665103235, 22.78020517657791755158535618604, 23.903371284641974179462038115749, 25.19677629400853146199432237620, 25.89249729973020582732093451176, 27.13023626706893155706560701848, 28.85821566603287548977270792786, 29.504004262289860379175022853375, 30.554870952471922096526282533645

Graph of the $Z$-function along the critical line