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Properties

Label 1-4033-4033.1062-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.161 - 0.986i$
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  + (0.766 + 0.642i)2-s + (0.396 − 0.918i)3-s + (0.173 + 0.984i)4-s + (0.973 + 0.230i)5-s + (0.893 − 0.448i)6-s + (−0.286 − 0.957i)7-s + (−0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (0.973 + 0.230i)12-s + (−0.286 − 0.957i)13-s + (0.396 − 0.918i)14-s + (0.597 − 0.802i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.396 − 0.918i)3-s + (0.173 + 0.984i)4-s + (0.973 + 0.230i)5-s + (0.893 − 0.448i)6-s + (−0.286 − 0.957i)7-s + (−0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (0.973 + 0.230i)12-s + (−0.286 − 0.957i)13-s + (0.396 − 0.918i)14-s + (0.597 − 0.802i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.161 - 0.986i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1062, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.161 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.233371359 - 1.897989462i\)
\(L(\frac12)\) \(\approx\) \(2.233371359 - 1.897989462i\)
\(L(1)\) \(\approx\) \(1.833245306 - 0.2320599785i\)
\(L(1)\) \(\approx\) \(1.833245306 - 0.2320599785i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (0.973 + 0.230i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (0.597 - 0.802i)T \)
13 \( 1 + (-0.286 - 0.957i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.993 + 0.116i)T \)
31 \( 1 + (0.973 - 0.230i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.0581 - 0.998i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (0.597 - 0.802i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.597 - 0.802i)T \)
79 \( 1 + (0.973 - 0.230i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (-0.0581 + 0.998i)T \)
97 \( 1 + (-0.286 - 0.957i)T \)
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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

−18.82535474241710775379929506078, −17.93104316427977971551551944689, −17.33837938058325298898953096022, −16.28075120668158911333295885268, −15.77344499486355945127168467806, −15.09726408641509854026520586122, −14.442176075119419267234296921, −13.86228085534509335686084026704, −13.3115911933231235276337405114, −12.43455728023392554219974058851, −11.7226129056719022687806383722, −11.22077750767605852113872804505, −10.16534656819248299669232203024, −9.58114116072477399808870864598, −9.292762876053269621402733526816, −8.67203012797768683356081212577, −7.20665168840212305278422135289, −6.36456325117957373543745084142, −5.71757159337580922215324800069, −4.93398084495271928740659736762, −4.46906441436342284909985790638, −3.63793012252720819212752106085, −2.48698474952986055283898814620, −2.3595852751623811509800794509, −1.403311030503336280980984528193, 0.51083801658283739055877673863, 1.696547650380069188107597611581, 2.44796257820914695783150461096, 3.41437090031731374414688345424, 3.753048213872057330286130247315, 5.05688962140387886208158161497, 5.79094674021778665485231311407, 6.51454794279895174772425255986, 6.762380366389836688827516962874, 7.81450698024934982107147301072, 8.20803647691000421603680138214, 9.15376327957372064630912078955, 9.948855958766829253617867167030, 10.8661290420701074605209308093, 11.65270561257126663248508864636, 12.4880982163144808649313083296, 13.12643790848427452166132727234, 13.68212639438782788818518444386, 13.97226402162530093426970406315, 14.706429134606698057175158782193, 15.36509254447402109980263526469, 16.51238247008991434935948092993, 16.8576053213479616719513851091, 17.70352306088248199136709566515, 17.97230309219011824500001045394

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