Properties

Degree 1
Conductor $ 11 \cdot 17 $
Sign $0.793 - 0.608i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.891 + 0.453i)2-s + (0.522 − 0.852i)3-s + (0.587 − 0.809i)4-s + (−0.760 + 0.649i)5-s + (−0.0784 + 0.996i)6-s + (0.852 − 0.522i)7-s + (−0.156 + 0.987i)8-s + (−0.453 − 0.891i)9-s + (0.382 − 0.923i)10-s + (−0.382 − 0.923i)12-s + (0.951 + 0.309i)13-s + (−0.522 + 0.852i)14-s + (0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (0.987 + 0.156i)19-s + ⋯
L(s,χ)  = 1  + (−0.891 + 0.453i)2-s + (0.522 − 0.852i)3-s + (0.587 − 0.809i)4-s + (−0.760 + 0.649i)5-s + (−0.0784 + 0.996i)6-s + (0.852 − 0.522i)7-s + (−0.156 + 0.987i)8-s + (−0.453 − 0.891i)9-s + (0.382 − 0.923i)10-s + (−0.382 − 0.923i)12-s + (0.951 + 0.309i)13-s + (−0.522 + 0.852i)14-s + (0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (0.987 + 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.793 - 0.608i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.793 - 0.608i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $0.793 - 0.608i$
motivic weight  =  \(0\)
character  :  $\chi_{187} (96, \cdot )$
Sato-Tate  :  $\mu(80)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 187,\ (0:\ ),\ 0.793 - 0.608i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8328726192 - 0.2828145740i$
$L(\frac12,\chi)$  $\approx$  $0.8328726192 - 0.2828145740i$
$L(\chi,1)$  $\approx$  0.8303567696 - 0.1154698020i
$L(1,\chi)$  $\approx$  0.8303567696 - 0.1154698020i

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

−27.27112386162192958426244389005, −26.70500064344727972254838246375, −25.48155710049461894408458882473, −24.80018189439097790249688265967, −23.576328215394433563147686642695, −22.037843653520275971368751764198, −21.26779314846566322985297837863, −20.28473590201008861138562819111, −19.947464542098290469249743582884, −18.624804078098613666105245407666, −17.716456819079468111044778651784, −16.34518150705723875959300193370, −15.87340152725073333230480901678, −14.86412967970660942380632161894, −13.40210874379149493485251607029, −11.983002000998001822833583990788, −11.28076614654101734830791847497, −10.209590431100108944966681792358, −8.93067595893639055916985881303, −8.43998524635119182053689730387, −7.51005552763959410058680194844, −5.41016374794470455705713843528, −4.11501385178357501679149831289, −3.06199407418943423539033695292, −1.47696320565713669985832204366, 1.024348331175245904547455588095, 2.396404282000741896706505598528, 3.979715088167704076130411763927, 5.95562653826964515433679932329, 7.02829917217433915851671343833, 7.85619580380082028078327484851, 8.45803987135461802367155083725, 9.910765017257300453235932851312, 11.22296412887815046937653788155, 11.80422086217717263555368819930, 13.65061273859974028516344197798, 14.41758083169387586753478492410, 15.30901173841057879302102927391, 16.42223073120254677106351531572, 17.74710393188082332414029500264, 18.34690792962272176120792017473, 19.12998844747008636643848249, 20.13136664489225762942963223112, 20.77479308336323716385282886633, 22.726861961033890643954600713, 23.730006186315006516810140622, 24.165031935566076432642463661752, 25.28174282261426233511134125068, 26.28299851140899159674576479660, 26.76403467300760817098039143289

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