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Properties

Label 1-43-43.40-r0-0-0
Degree $1$
Conductor $43$
Sign $0.0269 + 0.999i$
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  + (0.623 + 0.781i)2-s + (−0.988 − 0.149i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.900 + 0.433i)8-s + (0.955 + 0.294i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (0.365 − 0.930i)12-s + (0.0747 − 0.997i)13-s + (−0.988 + 0.149i)14-s + (−0.733 − 0.680i)15-s + (−0.900 − 0.433i)16-s + (0.826 − 0.563i)17-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.988 − 0.149i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.900 + 0.433i)8-s + (0.955 + 0.294i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (0.365 − 0.930i)12-s + (0.0747 − 0.997i)13-s + (−0.988 + 0.149i)14-s + (−0.733 − 0.680i)15-s + (−0.900 − 0.433i)16-s + (0.826 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.0269 + 0.999i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.0269 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6312039544 + 0.6144309601i\)
\(L(\frac12)\) \(\approx\) \(0.6312039544 + 0.6144309601i\)
\(L(1)\) \(\approx\) \(0.8963836628 + 0.5413745073i\)
\(L(1)\) \(\approx\) \(0.8963836628 + 0.5413745073i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (-0.988 - 0.149i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (0.955 - 0.294i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (-0.988 + 0.149i)T \)
31 \( 1 + (0.365 - 0.930i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (0.955 - 0.294i)T \)
71 \( 1 + (-0.733 - 0.680i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.988 - 0.149i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−33.83443255216844961805239608183, −33.07207425737266124533561085621, −32.26966289103927727674855613834, −30.591858199093243984543739763277, −29.4513173978352439234541552820, −28.72274853850344287010741858458, −27.89235228203764582884054022991, −26.2160416237976527768134634512, −24.37033135275627238924523445308, −23.35488766571014158893067784245, −22.38136425387917973928293862031, −21.20691775184445507511575705589, −20.275456532495087478952609483991, −18.59949840483354065370794629080, −17.28219229213822920255947176258, −16.10460503320502140744491174349, −14.19796169672790271091891479982, −12.94496537920759814447763876925, −11.98905908066448991388607816637, −10.34108839571527783695737858403, −9.672619435150973297878773662, −6.69329137984143319050504086237, −5.34041100712755420994037954811, −4.078543421435640068032895398476, −1.53587351594186967252160118865, 3.05843056762551056707845732391, 5.54015737798194062255898265238, 5.93239644544751770753859402002, 7.548762675684525216320346377912, 9.59790161289351084083442731052, 11.36076799283074629495149428170, 12.719761617234904940990205615637, 13.8191915697073123485313811102, 15.44909211982997308507344567467, 16.4653291768530653852702602739, 17.8074002397375762420920849133, 18.60167237013061575017267675416, 21.197198906423305052305517025536, 22.15928448097727831340765216564, 22.78271174028292973051466501783, 24.29734454020765955189420465767, 25.18604756030804506914996363736, 26.42281287977038442710131082158, 27.89290355951114601794328337008, 29.414392729542629154429544432383, 30.047304282845816050318641884270, 31.72438778648160853138277260000, 32.77477066640293680448594364133, 33.87733452247002868756759118861, 34.67057719693963064277728940225

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