Properties

Label 1-4003-4003.1049-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.763 - 0.645i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (−0.817 − 0.576i)2-s + (−0.619 − 0.785i)3-s + (0.336 + 0.941i)4-s + (0.197 + 0.980i)5-s + (0.0541 + 0.998i)6-s + (−0.999 + 0.0361i)7-s + (0.267 − 0.963i)8-s + (−0.232 + 0.972i)9-s + (0.403 − 0.915i)10-s + (−0.161 − 0.986i)11-s + (0.530 − 0.847i)12-s + (−0.856 − 0.515i)13-s + (0.837 + 0.546i)14-s + (0.647 − 0.762i)15-s + (−0.773 + 0.633i)16-s + (0.976 − 0.214i)17-s + ⋯
L(s)  = 1  + (−0.817 − 0.576i)2-s + (−0.619 − 0.785i)3-s + (0.336 + 0.941i)4-s + (0.197 + 0.980i)5-s + (0.0541 + 0.998i)6-s + (−0.999 + 0.0361i)7-s + (0.267 − 0.963i)8-s + (−0.232 + 0.972i)9-s + (0.403 − 0.915i)10-s + (−0.161 − 0.986i)11-s + (0.530 − 0.847i)12-s + (−0.856 − 0.515i)13-s + (0.837 + 0.546i)14-s + (0.647 − 0.762i)15-s + (−0.773 + 0.633i)16-s + (0.976 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.763 - 0.645i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3065542903 - 0.1121609393i\)
\(L(\frac12)\) \(\approx\) \(0.3065542903 - 0.1121609393i\)
\(L(1)\) \(\approx\) \(0.4097896217 - 0.1363996817i\)
\(L(1)\) \(\approx\) \(0.4097896217 - 0.1363996817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.817 - 0.576i)T \)
3 \( 1 + (-0.619 - 0.785i)T \)
5 \( 1 + (0.197 + 0.980i)T \)
7 \( 1 + (-0.999 + 0.0361i)T \)
11 \( 1 + (-0.161 - 0.986i)T \)
13 \( 1 + (-0.856 - 0.515i)T \)
17 \( 1 + (0.976 - 0.214i)T \)
19 \( 1 + (-0.674 - 0.738i)T \)
23 \( 1 + (-0.773 + 0.633i)T \)
29 \( 1 + (-0.674 + 0.738i)T \)
31 \( 1 + (-0.983 - 0.179i)T \)
37 \( 1 + (-0.674 + 0.738i)T \)
41 \( 1 + (-0.161 + 0.986i)T \)
43 \( 1 + (0.989 + 0.143i)T \)
47 \( 1 + (-0.773 - 0.633i)T \)
53 \( 1 + (-0.773 - 0.633i)T \)
59 \( 1 + (0.700 + 0.713i)T \)
61 \( 1 + (-0.891 + 0.452i)T \)
67 \( 1 + (0.837 - 0.546i)T \)
71 \( 1 + (-0.619 - 0.785i)T \)
73 \( 1 + (0.796 - 0.605i)T \)
79 \( 1 + (-0.891 - 0.452i)T \)
83 \( 1 + (-0.817 - 0.576i)T \)
89 \( 1 + (0.837 + 0.546i)T \)
97 \( 1 + (0.837 - 0.546i)T \)
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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

−18.505736258290753159958670066209, −17.449977406023396498231261322377, −17.13128588948080315703132567279, −16.610063583801076216142180773834, −15.97425830680319681592295018164, −15.5715206774262941333403601847, −14.56629339857786556446330865097, −14.197345575264248391246495631857, −12.677674016283202242986105937754, −12.49870641069014425364624297140, −11.68469165772821820686206927171, −10.56850611703857967711461922453, −10.07255260978079196479678597989, −9.51980786629840575437868474213, −9.14192388414073199305822803790, −8.166758226372402234462811327433, −7.35111440330636852630321504647, −6.56923745736372789193887483408, −5.77020265907218580239183492819, −5.36360724103652001464312778190, −4.412657686511457795058395152780, −3.83540152770583130763197204183, −2.38596119140423098945364763581, −1.58875193914771319424674654224, −0.33619791760661306506237029120, 0.37198674973159351749439053835, 1.548074779244328464441740619768, 2.42654808868521736850122381338, 3.083613621302669652631871546472, 3.61945923711185276271716132153, 5.119133424441198330402981272717, 6.00417613891682954182227946839, 6.52379555046216125699965977328, 7.37918146926601294915573892591, 7.68742161849975849088544418436, 8.6992200772362786767638471921, 9.61294420059742090141363544715, 10.217232524384688772777404954295, 10.79578833016764870924686566236, 11.4613637242805603241965761627, 12.08227055546512075239232682205, 12.886689628573155268215893093686, 13.305841049922428415035883740790, 14.13756753944858201273641343021, 15.09604219344713124931269448288, 16.03309314415151015276485788070, 16.58411142288410773041159285214, 17.18804258795333613604076473477, 17.91555978090638959586610252024, 18.4882893220405539322341219188

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