Properties

Degree 1
Conductor 43
Sign $0.0269 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.623 − 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.0269 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (14, \cdot )$
Sato-Tate  :  $\mu(21)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 43,\ (0:\ ),\ 0.0269 - 0.999i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6312039544 - 0.6144309601i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6312039544 - 0.6144309601i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8963836628 - 0.5413745073i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8963836628 - 0.5413745073i\)

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

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

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