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Properties

Label 1-4033-4033.36-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.509 - 0.860i$
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  + (−0.939 − 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 + 0.642i)4-s + (0.835 − 0.549i)5-s + (−0.993 − 0.116i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.973 + 0.230i)10-s + (−0.973 − 0.230i)11-s + (0.893 + 0.448i)12-s + (−0.0581 − 0.998i)13-s + (−0.286 + 0.957i)14-s + (0.686 − 0.727i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.973 − 0.230i)3-s + (0.766 + 0.642i)4-s + (0.835 − 0.549i)5-s + (−0.993 − 0.116i)6-s + (−0.0581 − 0.998i)7-s + (−0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.973 + 0.230i)10-s + (−0.973 − 0.230i)11-s + (0.893 + 0.448i)12-s + (−0.0581 − 0.998i)13-s + (−0.286 + 0.957i)14-s + (0.686 − 0.727i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.509 - 0.860i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.509 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027191450 - 1.802194932i\)
\(L(\frac12)\) \(\approx\) \(1.027191450 - 1.802194932i\)
\(L(1)\) \(\approx\) \(1.031409586 - 0.6156020088i\)
\(L(1)\) \(\approx\) \(1.031409586 - 0.6156020088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.973 - 0.230i)T \)
5 \( 1 + (0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (-0.973 - 0.230i)T \)
13 \( 1 + (-0.0581 - 0.998i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.686 + 0.727i)T \)
31 \( 1 + (0.0581 - 0.998i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.396 + 0.918i)T \)
53 \( 1 + (0.835 - 0.549i)T \)
59 \( 1 + (-0.686 + 0.727i)T \)
61 \( 1 + (-0.396 - 0.918i)T \)
67 \( 1 + (0.0581 - 0.998i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (0.893 - 0.448i)T \)
83 \( 1 + (-0.286 + 0.957i)T \)
89 \( 1 + (0.993 + 0.116i)T \)
97 \( 1 + (-0.893 - 0.448i)T \)
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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

−18.594073678131953992729570175163, −18.26328757363299998116860829671, −17.5678791525716205621561785215, −16.392799556882734560737869419873, −16.11460287865193589490636226476, −15.27490332927389893934076120804, −14.757992432140553753117525317315, −14.08210767884380256325251463189, −13.56578004852121765136417417145, −12.49434801089983995619422961155, −11.72980869172436264201109036858, −10.80049502757541999150951115270, −10.12083238237303144245294091292, −9.58200998029091419720215941008, −9.055148417087651583081881067818, −8.42646195956329540993623794167, −7.5519708002633215728464037461, −6.99892644674415201313846475166, −6.19574460571170318537571099393, −5.34075743436902746806306871559, −4.714638081872904110463764819570, −3.15983421670601576641812353184, −2.62063046645709093670143774817, −2.139018919362353973969381421159, −1.23207272688749737199903209325, 0.74254105694949422211625509190, 1.234068023525037745033639862790, 2.21706595963973884146099236616, 2.93959546280096436232975366839, 3.55189882321216537405795776694, 4.60421264573554220305174434871, 5.61317714927934402956836506615, 6.51705359500562402179282593657, 7.42758037816568160482261920561, 7.843479794908114487697888569310, 8.49694259568755780512613932001, 9.32343464972788225651377710973, 9.7848916797652093910127197162, 10.57208422088754916451457262032, 10.90621058752974717909287994916, 12.30039948256242530808223437328, 12.87765560986505937087587522995, 13.32211037451457023753417741599, 13.936002854565065994201664283924, 14.95036651414286823544551213341, 15.65592161831073266758878963874, 16.27324062057142502578230978085, 17.11585577843892100766868842219, 17.701815477013574542182251067777, 18.109196471124576563072496944578

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