Properties

Degree 1
Conductor 43
Sign $0.597 + 0.802i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.900 + 0.433i)2-s + (−0.900 − 0.433i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + 6-s + 7-s + (−0.222 + 0.974i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)11-s + (−0.900 + 0.433i)12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (0.623 − 0.781i)15-s + (−0.222 − 0.974i)16-s + (−0.222 − 0.974i)17-s + ⋯
L(s,χ)  = 1  + (−0.900 + 0.433i)2-s + (−0.900 − 0.433i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + 6-s + 7-s + (−0.222 + 0.974i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)11-s + (−0.900 + 0.433i)12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (0.623 − 0.781i)15-s + (−0.222 − 0.974i)16-s + (−0.222 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.597 + 0.802i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.597 + 0.802i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.597 + 0.802i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (16, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 43,\ (0:\ ),\ 0.597 + 0.802i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4152215619 + 0.2085424666i$
$L(\frac12,\chi)$  $\approx$  $0.4152215619 + 0.2085424666i$
$L(\chi,1)$  $\approx$  0.5613909332 + 0.1581272564i
$L(1,\chi)$  $\approx$  0.5613909332 + 0.1581272564i

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

−34.830839270352771682455547601908, −33.51970979043462086013191035728, −32.32090830268748033079851962266, −30.634795675666500527967748733131, −29.431111121911921877309941149060, −28.375082440315625314855334089016, −27.4833391642819479334251604619, −26.88538052332559775639288967873, −24.87332471278674974409182461024, −24.00945237222921162967163059673, −22.20115429299307847950876433408, −21.04759072768642403842667531966, −20.15726368574614744027728717503, −18.50928163086419443206312797220, −17.21384025293498022081297615555, −16.64508744266810447569243930925, −15.20516980901015282925606077744, −12.7298555499421654173738171643, −11.6440721375933827781062385030, −10.62609111873389419215973619354, −9.08398404337360924026337904047, −7.84044438713392466421261084333, −5.727112068802239035186642878636, −4.01727162665472089953963799868, −1.17479632412458385843176526894, 1.84512869148218032820911053930, 5.03791844876800415700544303229, 6.81982426699709969639381186048, 7.44760790660376124287900706326, 9.48732126479446889029488785664, 11.12299415583154307451661778804, 11.64821388538206973369449345088, 14.11641067399177898707061204688, 15.32643353516095096499418891954, 16.826636237187544073811127070914, 17.899258200480923597692937318002, 18.579011158971130670977685884627, 19.960670815916084932057330871771, 21.83811370158146224971419154778, 23.20512789483518870216534797840, 24.155747004221840157075836616619, 25.33443231223865981230274623968, 26.836513030362707587569294256846, 27.5975074002512375305779507641, 28.79379029800569195164717771932, 29.93416632027803122208569412359, 30.95600964529780835294973304612, 33.38981946261102568017618718160, 33.7109736377431593966944225065, 34.7981327004451202686681839807

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