Properties

Degree 1
Conductor $ 7 \cdot 29 $
Sign $-0.879 + 0.476i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.563 + 0.826i)2-s + (−0.149 + 0.988i)3-s + (−0.365 + 0.930i)4-s + (0.826 − 0.563i)5-s + (−0.900 + 0.433i)6-s + (−0.974 + 0.222i)8-s + (−0.955 − 0.294i)9-s + (0.930 + 0.365i)10-s + (0.294 + 0.955i)11-s + (−0.866 − 0.5i)12-s + (−0.222 + 0.974i)13-s + (0.433 + 0.900i)15-s + (−0.733 − 0.680i)16-s + (−0.866 + 0.5i)17-s + (−0.294 − 0.955i)18-s + (0.149 + 0.988i)19-s + ⋯
L(s,χ)  = 1  + (0.563 + 0.826i)2-s + (−0.149 + 0.988i)3-s + (−0.365 + 0.930i)4-s + (0.826 − 0.563i)5-s + (−0.900 + 0.433i)6-s + (−0.974 + 0.222i)8-s + (−0.955 − 0.294i)9-s + (0.930 + 0.365i)10-s + (0.294 + 0.955i)11-s + (−0.866 − 0.5i)12-s + (−0.222 + 0.974i)13-s + (0.433 + 0.900i)15-s + (−0.733 − 0.680i)16-s + (−0.866 + 0.5i)17-s + (−0.294 − 0.955i)18-s + (0.149 + 0.988i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.879 + 0.476i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.879 + 0.476i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(203\)    =    \(7 \cdot 29\)
\( \varepsilon \)  =  $-0.879 + 0.476i$
motivic weight  =  \(0\)
character  :  $\chi_{203} (10, \cdot )$
Sato-Tate  :  $\mu(84)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 203,\ (0:\ ),\ -0.879 + 0.476i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3571477396 + 1.408297936i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3571477396 + 1.408297936i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8719534520 + 0.9976128491i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8719534520 + 0.9976128491i\)

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

−26.48079314483849288140682808236, −25.17805834837046360424039665359, −24.47159881241804437323097818131, −23.55118476547045004183658424609, −22.37317107252531773460110193524, −22.03752003326094231431204017532, −20.75215875631424406873386121414, −19.68142232582224895229800085705, −18.97081363936760834216350682657, −17.92830370520379832334354579586, −17.39067618332064439819169911738, −15.53886371545004894403042692303, −14.324850958824410211219390111, −13.52014323120287687083171398515, −13.022835467749475503577703216270, −11.613014494358927400439953793546, −11.01699178222319199481734642237, −9.78384371893374164278035770760, −8.591271044850995943145798304918, −6.96125303366046694503979951596, −6.02670010907332333227448876781, −5.12268486542587015276387918149, −3.17401534527974643166156178688, −2.39769391304034161144999467395, −0.985786471473283051118638839093, 2.30316768246074009538404377632, 4.08391966539744094222880931717, 4.6912762256867113351826231006, 5.83707971225424379444813369268, 6.76624610363551743742490472859, 8.43812909391691715451972501312, 9.26031115189328679237165019333, 10.2064797881641613481545820769, 11.79024040292180993212874904599, 12.708582922187843534896233226818, 13.96151126972883404677247170704, 14.6710680166138470960355344222, 15.67666893464018957378510291004, 16.72858770077201067510585473386, 17.14283411893653559125552508282, 18.21427968379167037452683021110, 20.07825376193348016566206004397, 20.94509977098082751593565945367, 21.67550198926570716794652790421, 22.47380972721176865468323721948, 23.37959057340266667448691674871, 24.55872894217027673961710520131, 25.28810246730183841474473301475, 26.26176503617981613110971076344, 26.896833814605712772672998420689

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