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Properties

Label 1-4033-4033.3929-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.538 - 0.842i$
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  + 2-s + (−0.286 − 0.957i)3-s + 4-s + (0.893 − 0.448i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s + 8-s + (−0.835 + 0.549i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (−0.686 − 0.727i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.286 − 0.957i)3-s + 4-s + (0.893 − 0.448i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s + 8-s + (−0.835 + 0.549i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (−0.686 − 0.727i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.538 - 0.842i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.538 - 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.816755687 - 3.319330431i\)
\(L(\frac12)\) \(\approx\) \(1.816755687 - 3.319330431i\)
\(L(1)\) \(\approx\) \(1.787030553 - 1.085082574i\)
\(L(1)\) \(\approx\) \(1.787030553 - 1.085082574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (0.893 - 0.448i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (0.893 - 0.448i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.835 - 0.549i)T \)
31 \( 1 + (0.597 - 0.802i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.286 + 0.957i)T \)
53 \( 1 + (0.973 + 0.230i)T \)
59 \( 1 + (-0.686 - 0.727i)T \)
61 \( 1 + (0.396 - 0.918i)T \)
67 \( 1 + (-0.286 + 0.957i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (-0.0581 + 0.998i)T \)
97 \( 1 + (0.597 + 0.802i)T \)
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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

−18.71840556217461204589831443848, −17.95922495580955871076231903898, −16.96965157459346783065972157371, −16.56270834468773238002030197030, −15.88070965879845523773627245016, −15.2820026856353355954597939325, −14.59928246919551821761097383520, −13.896883841536528707585758934494, −13.51013707056229802391991035987, −12.49339366224350416725526133538, −11.8277602999289185311124459271, −11.236125640239183533367374132567, −10.39362258324418589946834416145, −9.99188431173445863092722971319, −9.056179313148113729133829296595, −8.53316365804825550705138513949, −7.0319188040895563353334079279, −6.43003285889445164482322402800, −5.75004959499618348501708917047, −5.607278658119996929026504959887, −4.35602604324061839635234207558, −3.52852783834754109159381878278, −3.33903749187639349892309378625, −2.18352792812358307728718289689, −1.392496082374393998914053002515, 0.73792080787540588376486459531, 1.446806445418798393353542230265, 2.340592853766362668917881767956, 3.013958906206617829849636446683, 4.02902016318279139621023276742, 4.78479646260249727393848312065, 5.73250314566367729961544140771, 6.18748493455862328540956943593, 6.75994361951360107688589402341, 7.41142388257499439134177896076, 8.304640129331578521738198355087, 9.375051798091185279669528483196, 9.94375226351597350620523826018, 10.94921275011472181743040735299, 11.567414959024283989550579391218, 12.25777925923859400683812640273, 13.00774373382274929747037541513, 13.37776819827388317867558062669, 13.874517918272338213376078876050, 14.46251637973098823166989739431, 15.6263121333911642553653194860, 16.176133310113883443573697445112, 16.90821983781953802602606797522, 17.42557846779854109855875951248, 18.1544367722361831614759196440

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