Properties

Degree 1
Conductor 193
Sign $-0.461 + 0.887i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.831 + 0.555i)2-s + (0.923 + 0.382i)3-s + (0.382 − 0.923i)4-s + (0.956 − 0.290i)5-s + (−0.980 + 0.195i)6-s + (0.707 + 0.707i)7-s + (0.195 + 0.980i)8-s + (0.707 + 0.707i)9-s + (−0.634 + 0.773i)10-s + (−0.881 + 0.471i)11-s + (0.707 − 0.707i)12-s + (−0.773 + 0.634i)13-s + (−0.980 − 0.195i)14-s + (0.995 + 0.0980i)15-s + (−0.707 − 0.707i)16-s + (−0.956 − 0.290i)17-s + ⋯
L(s,χ)  = 1  + (−0.831 + 0.555i)2-s + (0.923 + 0.382i)3-s + (0.382 − 0.923i)4-s + (0.956 − 0.290i)5-s + (−0.980 + 0.195i)6-s + (0.707 + 0.707i)7-s + (0.195 + 0.980i)8-s + (0.707 + 0.707i)9-s + (−0.634 + 0.773i)10-s + (−0.881 + 0.471i)11-s + (0.707 − 0.707i)12-s + (−0.773 + 0.634i)13-s + (−0.980 − 0.195i)14-s + (0.995 + 0.0980i)15-s + (−0.707 − 0.707i)16-s + (−0.956 − 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.461 + 0.887i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.461 + 0.887i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $-0.461 + 0.887i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (11, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 193,\ (1:\ ),\ -0.461 + 0.887i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9836422782 + 1.620974234i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9836422782 + 1.620974234i\)
\(L(\chi,1)\)  \(\approx\)  \(1.007495341 + 0.6028300774i\)
\(L(1,\chi)\)  \(\approx\)  \(1.007495341 + 0.6028300774i\)

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

−26.32087370787715906843845862368, −26.024086833404096359948588302559, −24.62538088221439288866518825979, −24.22931218678197412886695445876, −22.33504818595055700139328110528, −21.32748077168762953706098603719, −20.619387919780295867002601408329, −19.84073298389227532396938586383, −18.81197733231765069770396603935, −17.77459806412808960707812394912, −17.44149348696155806402209321785, −15.854783342974668923240166952271, −14.65331246105330798793169919454, −13.46782108973462455107793762474, −12.98311010719430164153765016629, −11.37610911079479557021011002694, −10.32130998173061371015336023153, −9.58074508458073249754053687914, −8.31206503535668233529453058919, −7.624578265573942829384408802962, −6.440490829691871431211575794641, −4.48035172123392693793580590009, −2.83878140384693991484726433853, −2.17120878947742185192138990392, −0.73599115478638206752600319946, 1.79225919631185230442506848734, 2.39834120328313444123401458409, 4.71902015265811565541374907044, 5.5577083984028894023590855249, 7.11704555699770306062664253480, 8.17615832465547377772549265, 9.07740657386640855873579262088, 9.80009915717342519572228579875, 10.769361813430441565577704665612, 12.398162875592194285826710976578, 13.90269829708816733040820368310, 14.4774431228733723192019960934, 15.54886882209890502653539865384, 16.3413995509414450557798248154, 17.69889576319928834441541703441, 18.22798611826604377295475910577, 19.3814530376565490206583813149, 20.438802062680674273259599203458, 21.13589271903536199943012390326, 22.1048897842628798510518717529, 23.854225001187602715287807256748, 24.69025254101329970613993046955, 25.2372873172840734438101709588, 26.16699379028544153025609862741, 26.882059763628157622858422897298

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