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Properties

Label 1-4033-4033.882-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.417 - 0.908i$
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 + (−0.549 + 0.835i)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 + (0.230 − 0.973i)10-s + (−0.230 − 0.973i)11-s + (−0.893 + 0.448i)12-s + (−0.0581 + 0.998i)13-s + (−0.286 − 0.957i)14-s + (0.727 − 0.686i)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 + (−0.549 + 0.835i)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 + (0.230 − 0.973i)10-s + (−0.230 − 0.973i)11-s + (−0.893 + 0.448i)12-s + (−0.0581 + 0.998i)13-s + (−0.286 − 0.957i)14-s + (0.727 − 0.686i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (882, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.417 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04601211358 + 0.02950693390i\)
\(L(\frac12)\) \(\approx\) \(0.04601211358 + 0.02950693390i\)
\(L(1)\) \(\approx\) \(0.3749430098 + 0.1650948951i\)
\(L(1)\) \(\approx\) \(0.3749430098 + 0.1650948951i\)

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 \( 1 + (-0.549 + 0.835i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.230 - 0.973i)T \)
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 \( 1 + (0.727 + 0.686i)T \)
31 \( 1 + (-0.998 + 0.0581i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.918 - 0.396i)T \)
53 \( 1 + (-0.549 + 0.835i)T \)
59 \( 1 + (-0.686 - 0.727i)T \)
61 \( 1 + (-0.918 - 0.396i)T \)
67 \( 1 + (-0.998 + 0.0581i)T \)
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 \( 1 + (-0.116 - 0.993i)T \)
97 \( 1 + (-0.448 - 0.893i)T \)
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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

−18.165169411772575203667289454933, −17.95245205147183853855987054425, −17.31188022947126915015612187089, −16.61639580536572985754644114198, −16.050079600557846158981625713248, −15.6777907112511293316017703185, −14.82060498837211196244908482528, −13.43883811080696665304228092200, −12.89000546567322723344076200056, −12.17251224305729030993044623561, −11.72432311756625919625939659771, −10.87403277130857641476551418708, −10.394221204969232709362558389892, −9.60127336563626503159860447606, −9.2014044816914344409586641176, −7.839699386556508925317966522124, −7.65183193118138641217086082495, −6.8910311912724054727825354738, −6.00216041884891534602206164929, −4.91039551156342378098118570810, −4.46600793890562237297229104936, −3.60443731723654361657276136183, −2.61029238476481925613268582146, −1.36338286769496156604861405465, −0.72317404051297469350883561634, 0.03674959222189698103176310160, 1.426910807871684764306590628890, 2.08883362304435429168659009373, 3.11196172117467094939056973444, 4.04644327243047092477537845812, 5.29678708513480061314328763263, 5.90153143294845430026668577853, 6.35094987509062485036978592463, 7.158548656017613300387233629472, 7.79477671587259660322612137187, 8.533667822288521531608656252067, 9.31465393772709390486413654461, 10.15899323544030698310212349470, 10.88261323742118810846222906398, 11.25360731660960044995213177824, 12.06776053925803749587141799821, 12.37455804417193740054765575053, 13.82251733939343224455882459574, 14.338590406882407056325766818180, 15.35555123997263814217470127822, 15.74644959187289569525630563763, 16.35810632214687919635410164182, 16.94439856734812541708780446797, 17.902797445004973448584466126175, 18.30950785831834471931641704388

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