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Properties

Label 1-4009-4009.1016-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.260 - 0.965i$
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  + (0.508 − 0.860i)2-s + (−0.909 − 0.415i)3-s + (−0.482 − 0.875i)4-s + (−0.567 − 0.823i)5-s + (−0.820 + 0.571i)6-s + (0.936 − 0.351i)7-s + (−0.999 − 0.0299i)8-s + (0.654 + 0.756i)9-s + (−0.997 + 0.0697i)10-s + (−0.842 − 0.538i)11-s + (0.0747 + 0.997i)12-s + (−0.952 + 0.304i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.534 + 0.845i)16-s + (0.591 + 0.806i)17-s + ⋯
L(s)  = 1  + (0.508 − 0.860i)2-s + (−0.909 − 0.415i)3-s + (−0.482 − 0.875i)4-s + (−0.567 − 0.823i)5-s + (−0.820 + 0.571i)6-s + (0.936 − 0.351i)7-s + (−0.999 − 0.0299i)8-s + (0.654 + 0.756i)9-s + (−0.997 + 0.0697i)10-s + (−0.842 − 0.538i)11-s + (0.0747 + 0.997i)12-s + (−0.952 + 0.304i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.534 + 0.845i)16-s + (0.591 + 0.806i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.260 - 0.965i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.260 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6353423018 - 0.8292760224i\)
\(L(\frac12)\) \(\approx\) \(0.6353423018 - 0.8292760224i\)
\(L(1)\) \(\approx\) \(0.6196921696 - 0.5784740203i\)
\(L(1)\) \(\approx\) \(0.6196921696 - 0.5784740203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.508 - 0.860i)T \)
3 \( 1 + (-0.909 - 0.415i)T \)
5 \( 1 + (-0.567 - 0.823i)T \)
7 \( 1 + (0.936 - 0.351i)T \)
11 \( 1 + (-0.842 - 0.538i)T \)
13 \( 1 + (-0.952 + 0.304i)T \)
17 \( 1 + (0.591 + 0.806i)T \)
23 \( 1 + (0.848 + 0.529i)T \)
29 \( 1 + (-0.952 + 0.304i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (0.887 + 0.460i)T \)
41 \( 1 + (-0.863 + 0.504i)T \)
43 \( 1 + (-0.583 + 0.811i)T \)
47 \( 1 + (0.987 - 0.158i)T \)
53 \( 1 + (0.778 + 0.627i)T \)
59 \( 1 + (-0.00498 - 0.999i)T \)
61 \( 1 + (0.559 + 0.829i)T \)
67 \( 1 + (-0.318 - 0.947i)T \)
71 \( 1 + (-0.241 - 0.970i)T \)
73 \( 1 + (0.542 + 0.840i)T \)
79 \( 1 + (-0.784 - 0.619i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.726 - 0.687i)T \)
97 \( 1 + (0.0548 - 0.998i)T \)
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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

−18.34267653389985825061796152859, −17.97801639532659529524647661766, −17.20008580539811258370209167280, −16.61376996491234382780641292843, −15.8279755154040878871996023996, −15.17967734908040171193618665689, −14.859181595881636398468544517595, −14.268980079970051683373044420841, −13.168910973044469820027568537035, −12.40318151124644812374261589232, −11.89487681696980946434777491364, −11.252054178481717817562678272637, −10.49956645830085514777667484869, −9.75807186299163988359358105100, −8.834474245547827792643617451655, −7.88306254091728479165631623989, −7.19522427826964784908449590041, −7.00834258999331941061713855793, −5.661261432229348552135186128449, −5.34445203703310517420243073305, −4.662673612584764965896075977572, −3.95269199951171448496614462696, −2.99924366592742340317078330122, −2.21358308771071127502790488948, −0.47466236739200472206615177074, 0.61653483759701550206912197655, 1.414244308975560162413276242, 2.04633803849405581866710527069, 3.24407079103863227264074777582, 4.17595114664670422761070768019, 4.80124419355049029290150335946, 5.34535302108588953846783696819, 5.8665832419239450225152197676, 7.14380997927619711040039028119, 7.76257987446117483867097148931, 8.48244881452640441869464086774, 9.46181361456390606312631090815, 10.28163030890743866485879398281, 10.99213055035452711932977683172, 11.47415450410183859659228073935, 12.00137132003321526536079868743, 12.87652404783708023510748375976, 13.08657521332456558247484392720, 13.961255534699069024744364088335, 14.92542154240861502559246904592, 15.33777233911306785285182227788, 16.60983083505002223462042922714, 16.81116007032796039500837150572, 17.6501883836038975737736624824, 18.50924620564557219557568810210

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