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Properties

Label 1-4033-4033.4029-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.942 + 0.332i$
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 + (−0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + 5-s + (−0.939 − 0.342i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.173 + 0.984i)14-s + (−0.5 + 0.866i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + 5-s + (−0.939 − 0.342i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.173 + 0.984i)14-s + (−0.5 + 0.866i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.942 + 0.332i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (4029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.942 + 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324607653 + 0.2269953120i\)
\(L(\frac12)\) \(\approx\) \(1.324607653 + 0.2269953120i\)
\(L(1)\) \(\approx\) \(0.8777970768 + 0.5642364019i\)
\(L(1)\) \(\approx\) \(0.8777970768 + 0.5642364019i\)

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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−18.028549417526281796312267972776, −17.876348253836898603367923883735, −17.6003840619608381091583521527, −16.876438168482187192541974506990, −15.53092829798181389050860645265, −14.725732982390524459190594390631, −14.070367644005321174062292373267, −13.55221391301510702180233332266, −12.7343693293839357125240471704, −12.46417335635605611488607685034, −11.49076491216140671005745105039, −11.02020985464827592652056283476, −10.33296902719265842415147699274, −9.55401893705154286880468438808, −8.88112905117462124954623597397, −7.93156979960013818876908145447, −7.29137069362877872851607876674, −6.33161817673194149718902814849, −5.42308869631997495808625465312, −5.08942411049952574044765142632, −4.33239174468425181113185232684, −2.98596999933708677721243578038, −2.27886161693294395008597856661, −1.63928230905627294740218396337, −1.12088529285056674186705866413, 0.38800103142313794966796810404, 1.6538095939858148529786272383, 2.750198037649798097749411999152, 3.84099000470046186965003868789, 4.52694234669356641263519412039, 5.08033925226026616955145491958, 5.8860904428839739152340027643, 6.2250618935168151238108059631, 7.16452953099798259112982999191, 8.095849753005001059910158993272, 8.91477412514006666480148409339, 9.321285675161650844177665496092, 10.06168318201092675707229019720, 10.90289770771482881984324986331, 11.611589607297730100052561769310, 12.28731814199980713564064234230, 13.44571823447143686449035188972, 13.91801970862867641726837513677, 14.50602363943360068290949105885, 15.029175957637971568214288939982, 15.907138952063677001088701821569, 16.638546876060220106250612998561, 16.93068834314629251809993435812, 17.55351444325625226931201360810, 18.3993139461736328216647794808

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