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Properties

Label 1-4033-4033.484-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.928 - 0.371i$
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 + (−0.835 − 0.549i)3-s + 4-s + (−0.597 − 0.802i)5-s + (0.835 + 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.835 − 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.0581 + 0.998i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.835 − 0.549i)3-s + 4-s + (−0.597 − 0.802i)5-s + (0.835 + 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.835 − 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.0581 + 0.998i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.928 - 0.371i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.928 - 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5328419087 - 0.1027527945i\)
\(L(\frac12)\) \(\approx\) \(0.5328419087 - 0.1027527945i\)
\(L(1)\) \(\approx\) \(0.4645385128 - 0.1924062819i\)
\(L(1)\) \(\approx\) \(0.4645385128 - 0.1924062819i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.835 - 0.549i)T \)
5 \( 1 + (-0.597 - 0.802i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (0.597 - 0.802i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.396 + 0.918i)T \)
31 \( 1 + (0.286 - 0.957i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.835 + 0.549i)T \)
53 \( 1 + (0.893 - 0.448i)T \)
59 \( 1 + (0.0581 + 0.998i)T \)
61 \( 1 + (0.686 - 0.727i)T \)
67 \( 1 + (-0.835 + 0.549i)T \)
71 \( 1 + T \)
73 \( 1 + (0.396 + 0.918i)T \)
79 \( 1 + (0.0581 + 0.998i)T \)
83 \( 1 + (0.973 + 0.230i)T \)
89 \( 1 + (0.993 - 0.116i)T \)
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

−18.48138442684402830815428762736, −17.85307453285940552604205504030, −17.26947449095403173465870517673, −16.411442883943787453272511485002, −16.02962675456843167758160087709, −15.17936526949828614741426471508, −14.76630233182031698180136594888, −14.17384410161419555242459007187, −12.349772441900364373732716570407, −11.92831584826656692573819165981, −11.81377114051571099301077965287, −10.84325146899361719119390998752, −10.18868931043465330458540139474, −9.61755526454951193275295409976, −8.96632662564243236158607863482, −8.03187305952619984085730462646, −7.295186694573998453618173505366, −6.64193563141921866596318368226, −6.03791432402657410027233176734, −5.13455287241088403110401748887, −4.22945487602415283070673797763, −3.435781437095213030511556333788, −2.35827149988217128932782378136, −1.74690964204744056127364151704, −0.35842039148461165845825226774, 0.81530000234179214218619075503, 1.08341610815833454466266094452, 2.09051194191775156540229452998, 3.333067674116148537863633232, 4.179159530702522926641237262021, 5.134431290577634853907779109522, 5.84629061920123767561843997307, 6.63791795345850846588463413271, 7.45424706196552592207210311318, 7.936776130401295766300155389358, 8.43139829587139987750820679145, 9.48564814128207649076553910287, 10.16983234477863425985774458177, 11.07974635267208544239595532610, 11.31532350325988139765214987997, 12.07653311431662484081860108359, 12.81790183127458755461187781065, 13.36334395704859256243096193127, 14.49504679615018068425631144064, 15.22173157677426469280264515101, 16.198889863714037401305518460290, 16.51305143885561610580724331114, 17.21978746068200340177314102656, 17.530061647970811464168886085300, 18.276043177004871081369072162935

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