Properties

Label 1-43-43.12-r1-0-0
Degree $1$
Conductor $43$
Sign $0.987 - 0.159i$
Analytic cond. $4.62099$
Root an. cond. $4.62099$
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 + (−0.365 + 0.930i)3-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + (0.900 + 0.433i)8-s + (−0.733 − 0.680i)9-s + (0.826 + 0.563i)10-s + (−0.222 + 0.974i)11-s + (0.988 + 0.149i)12-s + (0.826 − 0.563i)13-s + (0.365 + 0.930i)14-s + (0.955 + 0.294i)15-s + (−0.900 + 0.433i)16-s + (0.0747 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.365 + 0.930i)3-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + (0.900 + 0.433i)8-s + (−0.733 − 0.680i)9-s + (0.826 + 0.563i)10-s + (−0.222 + 0.974i)11-s + (0.988 + 0.149i)12-s + (0.826 − 0.563i)13-s + (0.365 + 0.930i)14-s + (0.955 + 0.294i)15-s + (−0.900 + 0.433i)16-s + (0.0747 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8651435366 - 0.06948801848i\)
\(L(\frac12)\) \(\approx\) \(0.8651435366 - 0.06948801848i\)
\(L(1)\) \(\approx\) \(0.7217042178 + 0.1298897986i\)
\(L(1)\) \(\approx\) \(0.7217042178 + 0.1298897986i\)

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 \( 1 + (-0.365 + 0.930i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (0.733 - 0.680i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (-0.988 - 0.149i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (0.988 - 0.149i)T \)
67 \( 1 + (-0.733 + 0.680i)T \)
71 \( 1 + (-0.955 - 0.294i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
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

−34.86256969578300603120038348830, −33.7166331981691213843422078427, −31.26634314868370729338436227609, −30.75019460653250636441506799153, −29.599132459400913432967063974995, −28.70887704764327005570151571803, −27.51802825635334122334605009009, −26.21449532916947793130998054748, −25.09231047124840736732217712371, −23.5797988055065721053765380296, −22.20532197976015736770184812606, −21.24206345648156518775906895738, −19.37827030235949129315061426880, −18.587302004619019220488286445245, −17.94013459789881565051283653998, −16.37158444483301817492120608395, −14.32832533174625196141207671720, −12.91492664909831618456122632203, −11.50051511632705199824514461330, −10.8794585472712594171868841746, −8.78435852999417849195104907300, −7.56884869833070360760553074196, −5.93940046623998932229186052, −3.19215406777822849195728449990, −1.63070017337273275805964874948, 0.75464555091294477432823711777, 4.43150706577305845615998160846, 5.406330071032371610959356331455, 7.39921954067244541480225982384, 8.89056241174787244984262003096, 10.012313576869300424364004614477, 11.33712876187275470438067024178, 13.46383654725152758935925798567, 15.058727811266082430131000983121, 16.08745144086917331185193158152, 17.0662825661934945761851996770, 17.98950950929495313539834653172, 20.18101400095924006822799283807, 20.67169226621048199178024043559, 22.783130030088719240112433203244, 23.59314478340867033296330902822, 24.948018405352451938533337688334, 26.21127433799121185342648243674, 27.3428211341036078247051242308, 28.05099161708062220376992352501, 29.07652313595288026284423944071, 31.1684169778002132638842081940, 32.668757720052749219349546936745, 33.04182717058085161933402128704, 34.16315745483041110319265840908

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