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Properties

Label 1-4009-4009.1057-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.628 - 0.777i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
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  + (0.525 − 0.850i)2-s + (0.971 + 0.237i)3-s + (−0.447 − 0.894i)4-s + (0.971 − 0.237i)5-s + (0.712 − 0.701i)6-s + (−0.525 − 0.850i)7-s + (−0.995 − 0.0896i)8-s + (0.887 + 0.460i)9-s + (0.309 − 0.951i)10-s + (0.134 − 0.990i)11-s + (−0.222 − 0.974i)12-s + (0.599 − 0.800i)13-s − 14-s + 15-s + (−0.599 + 0.800i)16-s + (−0.193 + 0.981i)17-s + ⋯
L(s)  = 1  + (0.525 − 0.850i)2-s + (0.971 + 0.237i)3-s + (−0.447 − 0.894i)4-s + (0.971 − 0.237i)5-s + (0.712 − 0.701i)6-s + (−0.525 − 0.850i)7-s + (−0.995 − 0.0896i)8-s + (0.887 + 0.460i)9-s + (0.309 − 0.951i)10-s + (0.134 − 0.990i)11-s + (−0.222 − 0.974i)12-s + (0.599 − 0.800i)13-s − 14-s + 15-s + (−0.599 + 0.800i)16-s + (−0.193 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.628 - 0.777i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.628 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.775387164 - 3.719233185i\)
\(L(\frac12)\) \(\approx\) \(1.775387164 - 3.719233185i\)
\(L(1)\) \(\approx\) \(1.696750131 - 1.370295117i\)
\(L(1)\) \(\approx\) \(1.696750131 - 1.370295117i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.525 - 0.850i)T \)
3 \( 1 + (0.971 + 0.237i)T \)
5 \( 1 + (0.971 - 0.237i)T \)
7 \( 1 + (-0.525 - 0.850i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
13 \( 1 + (0.599 - 0.800i)T \)
17 \( 1 + (-0.193 + 0.981i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (0.992 + 0.119i)T \)
31 \( 1 + (0.955 + 0.294i)T \)
37 \( 1 + (0.925 + 0.379i)T \)
41 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.0448 + 0.998i)T \)
53 \( 1 + (0.550 + 0.834i)T \)
59 \( 1 + (0.873 + 0.486i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (0.826 + 0.563i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.420 + 0.907i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.646 - 0.762i)T \)
97 \( 1 + (-0.163 + 0.986i)T \)
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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

−18.472855502956471136300535268242, −18.03206329885244096044635584437, −17.52418968558781508116760247533, −16.41014700531166378997729120112, −15.90298198614842419269395066958, −15.16305855733563640399823390180, −14.65861381406237145354333349236, −13.96067079567593767178159364974, −13.39103326591076150065799421427, −12.95268088036454436106218594571, −12.109040620828200686165614987129, −11.46527035577660877330575129517, −9.906405528068118482134777494621, −9.54566752840356737128191701816, −9.01086229666260768527254520808, −8.26625494133979141612068980927, −7.3720533730809259483663563186, −6.659296017424375456282150897940, −6.30717143130758035503558179459, −5.32939340109870545747214660312, −4.552732743074788502222010179492, −3.72598930226277539966334131054, −2.69991695048479032972418061263, −2.44565337990923645908240058728, −1.32771071753772102737886025123, 0.90017743894680147293406987501, 1.35542016367412852523639600143, 2.65619407642904407192110542968, 2.87905776782862821301695051477, 3.90908591364287539911768089152, 4.34311448741020075676932221478, 5.386152956577234345379957214308, 6.18143879308030446637119788184, 6.72139246369311528396955593508, 8.13182568642911094570247673994, 8.625442535897437040486721768952, 9.32989943883824859186701115468, 10.20794663548882238844815699978, 10.435926106452747719966653824931, 11.07912503254110503504570962460, 12.45052352920377095106662522691, 12.896937107539822680756919259250, 13.51014668764025399303957012834, 13.96765346468078616979829357625, 14.45431024716658650763723688489, 15.39527038088701013642512404555, 16.04925974126390138505838502784, 16.893536972127439329697982669065, 17.670387550466753711246572287630, 18.5078380200559247250392428932

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