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Properties

Label 1-4033-4033.1798-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.567 + 0.823i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.173 + 0.984i)2-s + (0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.342 − 0.939i)5-s + (0.173 + 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s − 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.642 − 0.766i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.342 − 0.939i)5-s + (0.173 + 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s − 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.642 − 0.766i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.567 + 0.823i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1798, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.567 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9608418728 + 0.5050341441i\)
\(L(\frac12)\) \(\approx\) \(0.9608418728 + 0.5050341441i\)
\(L(1)\) \(\approx\) \(0.8762209677 + 0.1807674340i\)
\(L(1)\) \(\approx\) \(0.8762209677 + 0.1807674340i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.984 - 0.173i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (-0.642 + 0.766i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.342 - 0.939i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.342 - 0.939i)T \)
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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

−18.61485633007128436968222823902, −18.07957115552918433333906613305, −17.05001187805144813749379336555, −16.29801121374960389517718024360, −15.416699907954118832389570848259, −15.04010924239680274654420166318, −14.015013438740083704356823551855, −13.52010129853329157061112401664, −13.10920715065603566138047513250, −12.10017796241063823067558544878, −11.40457615881894174315918192304, −10.570222590500921237834086702577, −10.12110646194822204628709105146, −9.50801230120753305942213208475, −8.90647957876260395929697794645, −7.954427938919740003924371754747, −7.260076507428472921165478158791, −6.795098407424409173082041335659, −5.297110078193523259794648711117, −4.54116033223698469802977718180, −3.78920727739204309460325544422, −3.09760202441486162273034076503, −2.57730380154548147768927921428, −1.995415119002647374491088659152, −0.42396114414947642337148227998, 0.66826300897500836741151600673, 1.668552396685876470449161794353, 2.950195401404566499863806006279, 3.40813857851323794332991475108, 4.57161235893240027592768576198, 5.1213149388722503538371688139, 5.92493622102669139195410379390, 6.89570594313920990669038815690, 7.42853304854709855164421326177, 8.15476656489181115773513582387, 8.72732582350155461490972692955, 9.28496957980611495289108707035, 9.92016003251987137925762454856, 10.71765261708795907502909160391, 12.284314610969999381477850835985, 12.55621581641436672665115282479, 13.33579034341308929980242570337, 13.60813022590611402186922010658, 14.71273044982772036291227726034, 15.35012201105444575082559426210, 15.73147678601700650489723923715, 16.37523229043329439967164573600, 17.07079282592045548739911936047, 18.07062565568325885217081864729, 18.38303276616161270247350243907

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