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Properties

Label 1-4033-4033.3210-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.296 + 0.955i$
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 + (0.893 + 0.448i)3-s + (−0.5 + 0.866i)4-s + (0.993 + 0.116i)5-s + (0.0581 + 0.998i)6-s + (0.597 − 0.802i)7-s − 8-s + (0.597 + 0.802i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.835 + 0.549i)12-s + (0.993 + 0.116i)13-s + (0.993 + 0.116i)14-s + (0.835 + 0.549i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.893 + 0.448i)3-s + (−0.5 + 0.866i)4-s + (0.993 + 0.116i)5-s + (0.0581 + 0.998i)6-s + (0.597 − 0.802i)7-s − 8-s + (0.597 + 0.802i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.835 + 0.549i)12-s + (0.993 + 0.116i)13-s + (0.993 + 0.116i)14-s + (0.835 + 0.549i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.296 + 0.955i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3210, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.296 + 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.494786159 + 2.575139196i\)
\(L(\frac12)\) \(\approx\) \(3.494786159 + 2.575139196i\)
\(L(1)\) \(\approx\) \(1.959800281 + 1.129542153i\)
\(L(1)\) \(\approx\) \(1.959800281 + 1.129542153i\)

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 \( 1 + (0.893 + 0.448i)T \)
5 \( 1 + (0.993 + 0.116i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (0.396 - 0.918i)T \)
13 \( 1 + (0.993 + 0.116i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (0.686 - 0.727i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (0.893 + 0.448i)T \)
59 \( 1 + (0.835 + 0.549i)T \)
61 \( 1 + (0.286 - 0.957i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.993 + 0.116i)T \)
79 \( 1 + (0.0581 - 0.998i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (0.993 + 0.116i)T \)
97 \( 1 + (0.686 + 0.727i)T \)
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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

−18.24777265271005205758511558479, −17.89294147268657238311197057420, −17.53083973569006571727719741596, −16.01410620820989673783813236544, −15.27167900028396807389715054699, −14.69911411803519577132107005248, −14.1331732827777681893327582602, −13.450706206523511510175490264951, −13.012930367875673794148393516233, −12.16906759062204909877332699420, −11.78315397762611995415105974123, −10.64057773964336978150435247194, −10.06651388971211811798890313575, −9.321886670897140484090925406575, −8.67114974057044986210262660569, −8.295368628726753786280595658935, −6.8874628401935195906445707940, −6.26219722013166701196712030113, −5.59145992364391257171674937581, −4.602524848955417459513659092988, −4.01929761414467199859880689157, −3.00597707644152825104094595859, −2.17815846141205477467781577633, −1.845070240200383541710861432700, −1.149523018605736988303902637785, 0.97046060869474753525870808578, 2.13906555275372368604598634902, 2.85803362702765539318124087082, 3.81574551581054804091350445621, 4.37270996559082125351391555148, 4.98784455776930859365005646264, 6.17503630515785791496755979837, 6.451269563274706305682821611225, 7.39423101605073729165252641836, 8.37797499829920530572369327873, 8.62432039967689211326783314886, 9.32664648088017414213505652364, 10.33323775653621470191321570740, 10.86246348906099518367332904934, 11.745301684128040350227786011804, 13.06712049188171320495595724305, 13.457168964992810042614354607346, 13.83460453376091844377286064388, 14.47406753040557289283400397741, 15.014518717210946133657780289180, 15.94914025174249772071341932424, 16.368974710146519054073415910931, 17.197008121736900240853845644851, 17.705270573045730156080392338451, 18.45105715038197538764763846288

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