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Properties

Label 1-4033-4033.1067-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.999 + 0.0194i$
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  + (0.5 + 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.5 + 0.866i)4-s + (0.984 − 0.173i)5-s + (−0.939 − 0.342i)6-s + (0.173 + 0.984i)7-s − 8-s + (0.173 − 0.984i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 − 0.984i)12-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)14-s + (−0.642 + 0.766i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.5 + 0.866i)4-s + (0.984 − 0.173i)5-s + (−0.939 − 0.342i)6-s + (0.173 + 0.984i)7-s − 8-s + (0.173 − 0.984i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 − 0.984i)12-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)14-s + (−0.642 + 0.766i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.999 + 0.0194i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.999 + 0.0194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6883099921 + 0.006693857603i\)
\(L(\frac12)\) \(\approx\) \(0.6883099921 + 0.006693857603i\)
\(L(1)\) \(\approx\) \(0.7359816818 + 0.6032278469i\)
\(L(1)\) \(\approx\) \(0.7359816818 + 0.6032278469i\)

Euler product

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

−18.55938318231561033189748800364, −17.84147293599325983755671122568, −17.36006039548209630053341704998, −16.55546125501050537780393977233, −15.94180756220052740816448709830, −14.555602028850709770153728551219, −14.195984115999999166022808855872, −13.486346771117734240639178295396, −13.01201326113961090957642605091, −12.51878532394593784274952946175, −11.34309421156371798149464802758, −10.99500528494473811075181864608, −10.5584283794135297041698997328, −9.711492292492073640242717744945, −8.96769477389862070791349455311, −7.95989058290523784491677439858, −6.951031129059769050103984443091, −6.37459178424100394452638693936, −5.75098123317650932451176735456, −4.90774470577185022095510657839, −4.39204262768559038840579256236, −3.27861714010298600059280350263, −2.38440541958749223800281011998, −1.67334284868146608361960604210, −1.01866471248141505368090580904, 0.18447149226834020994072478818, 1.82524775630083843324498309200, 2.698995152132868295096696384004, 3.59097089776331948994651590480, 4.58640157115536695088519677171, 5.38326213751205080669937846441, 5.50001768936261835820131166929, 6.14961831069030358451237164848, 7.16342875705245110411195904108, 7.83225009796113144687773995036, 8.842852575377141799519595442329, 9.52123141107212132757950623028, 9.911414809245555946314947038865, 10.91582342828450716383147623030, 11.818510067328050085068382745474, 12.446143027499934095030714073504, 12.92609448346701923903228528280, 13.78382934151511586969111123026, 14.544969438454791166541191555549, 15.29770641341822264049186144940, 15.646369504922170901767792989071, 16.30018508813408567836379423005, 17.17450470843359466085362990481, 17.60346021330038809626425070819, 18.22943073456712725572354349711

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