Properties

Degree 1
Conductor $ 3^{2} \cdot 11 $
Sign $0.540 - 0.841i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s − 10-s + (0.104 − 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.309 − 0.951i)26-s + ⋯
L(s,χ)  = 1  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s − 10-s + (0.104 − 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.540 - 0.841i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.540 - 0.841i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.540 - 0.841i$
motivic weight  =  \(0\)
character  :  $\chi_{99} (68, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 99,\ (0:\ ),\ 0.540 - 0.841i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.372457454 - 0.7492157435i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.372457454 - 0.7492157435i\)
\(L(\chi,1)\)  \(\approx\)  \(1.457986860 - 0.5216298586i\)
\(L(1,\chi)\)  \(\approx\)  \(1.457986860 - 0.5216298586i\)

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

−30.62282474847942097137326176118, −29.464382126926348221192196262731, −28.05179031306223201531852114497, −26.77350928563547646984235799819, −26.11572373007363037647809127593, −24.5037747544452321890860069264, −23.96391704254669218061351230372, −22.96663612869947363872397739311, −21.96860606802351079176118665047, −20.854412401062811959847914634272, −19.88845302414345893070070608711, −18.47991044468459046599317412734, −17.14402193017599610913068535731, −16.01368626582483396812993727216, −14.97785534704634143942205007448, −14.20540724993512080584081555526, −12.9294980088268836953411690513, −11.45117071066658450419997897316, −11.128839744711962472617894656939, −8.753801995082465880514583906634, −7.52787934280713278660858022514, −6.640263354927876887378783781291, −4.85157422590167755604411061695, −4.02234633910117394629498728522, −2.37787210554348468543368013898, 1.59490774131691463076170592320, 3.408227302811940080530499094653, 4.59079751163166935581379553378, 5.67684582116985077009282715948, 7.406444391583994110570532678488, 8.59512809477493975985816476229, 10.47954463339311538498939215197, 11.44497718329914859056512889856, 12.38109977541732088964199034489, 13.454833373716247840764480379907, 14.95132392372732685411106376744, 15.39802690997146770071238257172, 16.868944900954785785154637348208, 18.3956993166240360298629517191, 19.635655395829960302985862689341, 20.45157172258619997024616844876, 21.378805395672378433981060430533, 22.59856114656484891556614983775, 23.55763942263706060518242233487, 24.33888334724459461470619774543, 25.28669370243764867200265106219, 27.16116746966147579971667598617, 27.80138593269626121478928343882, 28.877954516797474408002814322575, 30.146129907645598081256244202713

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