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Properties

Label 1-4009-4009.1080-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.860 + 0.509i$
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  + (−0.991 + 0.129i)2-s + (0.943 − 0.332i)3-s + (0.966 − 0.256i)4-s + (−0.999 + 0.00997i)5-s + (−0.892 + 0.451i)6-s + (0.887 + 0.460i)7-s + (−0.925 + 0.379i)8-s + (0.778 − 0.627i)9-s + (0.990 − 0.139i)10-s + (−0.447 + 0.894i)11-s + (0.826 − 0.563i)12-s + (0.638 + 0.769i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (0.868 − 0.495i)16-s + (−0.784 + 0.619i)17-s + ⋯
L(s)  = 1  + (−0.991 + 0.129i)2-s + (0.943 − 0.332i)3-s + (0.966 − 0.256i)4-s + (−0.999 + 0.00997i)5-s + (−0.892 + 0.451i)6-s + (0.887 + 0.460i)7-s + (−0.925 + 0.379i)8-s + (0.778 − 0.627i)9-s + (0.990 − 0.139i)10-s + (−0.447 + 0.894i)11-s + (0.826 − 0.563i)12-s + (0.638 + 0.769i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (0.868 − 0.495i)16-s + (−0.784 + 0.619i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1080, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473069273 + 0.4034761664i\)
\(L(\frac12)\) \(\approx\) \(1.473069273 + 0.4034761664i\)
\(L(1)\) \(\approx\) \(0.9544548406 + 0.07894944647i\)
\(L(1)\) \(\approx\) \(0.9544548406 + 0.07894944647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.991 + 0.129i)T \)
3 \( 1 + (0.943 - 0.332i)T \)
5 \( 1 + (-0.999 + 0.00997i)T \)
7 \( 1 + (0.887 + 0.460i)T \)
11 \( 1 + (-0.447 + 0.894i)T \)
13 \( 1 + (0.638 + 0.769i)T \)
17 \( 1 + (-0.784 + 0.619i)T \)
23 \( 1 + (0.559 - 0.829i)T \)
29 \( 1 + (0.346 - 0.937i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (-0.550 + 0.834i)T \)
41 \( 1 + (-0.0647 + 0.997i)T \)
43 \( 1 + (0.698 + 0.715i)T \)
47 \( 1 + (-0.517 - 0.855i)T \)
53 \( 1 + (0.905 - 0.424i)T \)
59 \( 1 + (-0.831 - 0.555i)T \)
61 \( 1 + (-0.615 - 0.788i)T \)
67 \( 1 + (0.878 - 0.478i)T \)
71 \( 1 + (0.0348 - 0.999i)T \)
73 \( 1 + (-0.797 - 0.603i)T \)
79 \( 1 + (0.739 + 0.672i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.909 - 0.415i)T \)
97 \( 1 + (0.654 + 0.756i)T \)
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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

−18.45290753216003566363831948359, −17.93191425439162431604096389412, −17.110733990942073698602864280093, −16.17621162447193907922757797668, −15.71661732461118812216696091762, −15.371761740379859653214325510109, −14.44276243441290268837329392078, −13.74826835960164462984034857741, −12.96373701997088039466361656786, −12.05251038762770905340103405228, −11.189919269070477207022301710824, −10.76990997117387280693101128286, −10.305799128952548540174748588979, −9.03967912279220858947592982152, −8.66551651829719533494025054970, −8.16432840013978504958565257262, −7.3642204845204225356948150245, −7.09652240376217510548774015947, −5.68725207546829912781801631414, −4.7777573920038349863738283916, −3.8517074548368683757390338999, −3.20932022957674081220059874951, −2.58505407135794217356245068820, −1.43327638343149141154636620655, −0.689922896723283387368140654321, 0.863943938540228862618772947028, 1.827277074406306312017609639036, 2.33915762386812659373008264802, 3.22959745083387563232942357077, 4.298287296115778203561409283589, 4.829404221527565425384730941821, 6.36695487108227132397303681308, 6.73163704085235090995287484004, 7.75629011428183448443007195091, 8.06140158674042519146094569483, 8.64986947067717991343878643637, 9.25535166656995258631828265239, 10.14427423534226169904988494243, 10.94406569093272175437483034457, 11.615351176607643714163364358, 12.20122176098755265692671919961, 12.964675760099516542491230133, 13.941146451452165926435078994571, 14.807260046512758789531031149407, 15.27939972648516700690755007180, 15.5439480455568712234550277643, 16.502785733130829397780044725816, 17.34155394045618104144145778000, 18.10134542864519382957660159937, 18.57339737012711598104690568963

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