Properties

Degree 1
Conductor $ 3^{2} \cdot 11 $
Sign $0.0111 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0111 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0111 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.0111 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{99} (4, \cdot )$
Sato-Tate  :  $\mu(15)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 99,\ (0:\ ),\ 0.0111 - 0.999i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3383705635 - 0.3421586541i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3383705635 - 0.3421586541i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5421413238 - 0.1874494048i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5421413238 - 0.1874494048i\)

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

−30.18135384181414628450367861978, −28.80938411406162238134476176803, −28.02570518670590097666227074242, −27.36287522162681849081905236205, −26.10369838730935656304341381871, −25.30705719896401581079038747039, −24.0796518870157794282012195258, −23.35929150603491656967713992391, −21.66986882104673682478083891439, −20.598036851793345528027205111001, −19.2437565025776471638702621232, −18.89992133137870086317936045716, −17.546752403462994876137257216856, −16.27312205905846535713366677969, −15.586508423190078970651880780, −14.53549063206877265234749878694, −12.47266570616066931687290804278, −11.630812375018585336041313917113, −10.44316026883232743802042251149, −8.89354041276842569541903398381, −8.299930895302125940279658015478, −6.88477439082133108542802630835, −5.59404224865048564322902139041, −3.62750843694828395923812951632, −1.78486359535053587161088633436, 0.688710905620312400505734838760, 2.88485814136192363869274916891, 4.19691046428554717619163619369, 6.486081704814932325732725336051, 7.58259039150894318913026881646, 8.460603608739471774266639478985, 10.03761980960323485016231052130, 10.94486271932952727915277098562, 11.93937269674484535110744264679, 13.34064413743798640333913989184, 15.033024671198363450739676015740, 16.02345814450995913637024774718, 16.96771577871973789143905913171, 18.17849182074494829636526195479, 19.17693739015679248385140705507, 20.12905594946311009562811615358, 20.83430975105193978182755319410, 22.55090204885744943176462996482, 23.516751078636817604284070025993, 24.686610434562015578167198684133, 25.96886827903973733576249689550, 26.72202950649130454240329422036, 27.604849893152228057223173421174, 28.434324323235136212625691760524, 30.005991147218985657097849748608

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