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Properties

Label 1-4033-4033.84-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.906 + 0.422i$
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 + (−0.993 − 0.116i)3-s + (0.173 − 0.984i)4-s + (−0.286 + 0.957i)5-s + (0.835 − 0.549i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.396 − 0.918i)10-s + (−0.396 + 0.918i)11-s + (−0.286 + 0.957i)12-s + (0.686 + 0.727i)13-s + (0.993 + 0.116i)14-s + (0.396 − 0.918i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.993 − 0.116i)3-s + (0.173 − 0.984i)4-s + (−0.286 + 0.957i)5-s + (0.835 − 0.549i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.396 − 0.918i)10-s + (−0.396 + 0.918i)11-s + (−0.286 + 0.957i)12-s + (0.686 + 0.727i)13-s + (0.993 + 0.116i)14-s + (0.396 − 0.918i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.906 + 0.422i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1215327688 + 0.5480333009i\)
\(L(\frac12)\) \(\approx\) \(0.1215327688 + 0.5480333009i\)
\(L(1)\) \(\approx\) \(0.4396178467 + 0.2446454456i\)
\(L(1)\) \(\approx\) \(0.4396178467 + 0.2446454456i\)

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 \( 1 + (-0.993 - 0.116i)T \)
5 \( 1 + (-0.286 + 0.957i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (-0.396 + 0.918i)T \)
13 \( 1 + (0.686 + 0.727i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.597 - 0.802i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.893 + 0.448i)T \)
53 \( 1 + (0.686 + 0.727i)T \)
59 \( 1 + (-0.396 + 0.918i)T \)
61 \( 1 + (-0.835 + 0.549i)T \)
67 \( 1 + (-0.973 - 0.230i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (0.286 + 0.957i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (0.893 + 0.448i)T \)
97 \( 1 + (-0.686 - 0.727i)T \)
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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

−18.14461842534017940891263984352, −17.58038619655178981963509323007, −16.75683213815500568791842594184, −16.32138332614576718870966460995, −15.79970370959886687978667344875, −15.27247559346247077631988727728, −13.68880186187130230124917703481, −12.90796697055769051763242141713, −12.6287501217555708473438786956, −11.954381393324154761184814951414, −11.25522715482908677877719079179, −10.63295120858615068202644987965, −10.00928859597793547939293262836, −8.96863925113191684996766360632, −8.80097458129339233561321718718, −7.86287520709198121313579925696, −7.004306663244574911759185159859, −6.14673007432166972617483783449, −5.35770880167414421896541724658, −4.81131219156438782675290287857, −3.51219227372429332151828392070, −3.2414027702394443290039518072, −1.93247023295723998268286178452, −0.91581616670135154589689638818, −0.38666938812059010274678574136, 0.87107336761394536678806758733, 1.74158211515790324532237211571, 2.79341167725300178490339492758, 4.05588231648331325003985173866, 4.53895554956269850932537033420, 5.73884736972252970139714147856, 6.40134750300985439856670752840, 6.66704565762550155025654577149, 7.65276237414041402774296619649, 7.831576510150591865598596125328, 9.24600664120575469931858863558, 9.95969380203369752726741493534, 10.41364379086719356869632821392, 10.96195002193990799273652907395, 11.65675473859772814381901859927, 12.5636633421274758239756126237, 13.33204332879087988093715523018, 14.14997742364673384001506685608, 14.9873663517301894574461072933, 15.40899444607272561875243239185, 16.382238443755834331308209309830, 16.65152669253106144985155887742, 17.3227095872748326433288711780, 18.188050884481443050835546548423, 18.49531466195581520067555700176

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