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Properties

Label 1-43-43.35-r0-0-0
Degree $1$
Conductor $43$
Sign $0.597 - 0.802i$
Analytic cond. $0.199691$
Root an. cond. $0.199691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.597 - 0.802i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.597 - 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4152215619 - 0.2085424666i\)
\(L(\frac12)\) \(\approx\) \(0.4152215619 - 0.2085424666i\)
\(L(1)\) \(\approx\) \(0.5613909332 - 0.1581272564i\)
\(L(1)\) \(\approx\) \(0.5613909332 - 0.1581272564i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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