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Properties

Label 1-4009-4009.10-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.766 - 0.642i$
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  + (−0.882 − 0.469i)2-s + (−0.997 − 0.0697i)3-s + (0.559 + 0.829i)4-s + (−0.241 − 0.970i)5-s + (0.848 + 0.529i)6-s + (−0.309 − 0.951i)7-s + (−0.104 − 0.994i)8-s + (0.990 + 0.139i)9-s + (−0.241 + 0.970i)10-s + (0.913 − 0.406i)11-s + (−0.5 − 0.866i)12-s + (0.882 − 0.469i)13-s + (−0.173 + 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.374 + 0.927i)16-s + (−0.990 + 0.139i)17-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (−0.997 − 0.0697i)3-s + (0.559 + 0.829i)4-s + (−0.241 − 0.970i)5-s + (0.848 + 0.529i)6-s + (−0.309 − 0.951i)7-s + (−0.104 − 0.994i)8-s + (0.990 + 0.139i)9-s + (−0.241 + 0.970i)10-s + (0.913 − 0.406i)11-s + (−0.5 − 0.866i)12-s + (0.882 − 0.469i)13-s + (−0.173 + 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.374 + 0.927i)16-s + (−0.990 + 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2309693429 - 0.6350010942i\)
\(L(\frac12)\) \(\approx\) \(0.2309693429 - 0.6350010942i\)
\(L(1)\) \(\approx\) \(0.4679143692 - 0.2801009085i\)
\(L(1)\) \(\approx\) \(0.4679143692 - 0.2801009085i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.882 - 0.469i)T \)
3 \( 1 + (-0.997 - 0.0697i)T \)
5 \( 1 + (-0.241 - 0.970i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.882 - 0.469i)T \)
17 \( 1 + (-0.990 + 0.139i)T \)
23 \( 1 + (0.374 + 0.927i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
31 \( 1 + T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.848 + 0.529i)T \)
53 \( 1 + (0.719 + 0.694i)T \)
59 \( 1 + (0.719 - 0.694i)T \)
61 \( 1 + (-0.961 + 0.275i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.997 + 0.0697i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.719 - 0.694i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.882 + 0.469i)T \)
97 \( 1 + (0.559 - 0.829i)T \)
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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

−18.45336697788322168202305993215, −18.12866181206700137242549827246, −17.518235092810236168941234321898, −16.71133046696544459233513399213, −16.084796944689522757296277631925, −15.474609730256591542437004223653, −15.0010483837142704540185230843, −14.26614078653103302784665946330, −13.268954435798034208918473709470, −12.24198291153636352837700524268, −11.61644655004349113410705425376, −11.12723406235395209703709642419, −10.55435617211985501815715706083, −9.65999165691902658393274359319, −9.13836857692564592029136642920, −8.37903303809542107907564702188, −7.27204890146053124980108214587, −6.79413078325132474829751621088, −6.15907550823875301316645334357, −5.80541530514875563344144419058, −4.607147745037591939893254695678, −3.87471888065008309135260801897, −2.56576868042687364909486698551, −1.93729529988637515268644046465, −0.80672108189124708801060470824, 0.45114173174465314494770868503, 1.13956856884199048493489073178, 1.6412778613974287591605871622, 3.15839074350150843301798712859, 4.03135179420560501072787314010, 4.3853701332983519788412007791, 5.63750314205597733544475818728, 6.33269644844478946802208964, 7.07784556125542709834948308028, 7.736063922745808294868282007603, 8.61788848211711989578676365640, 9.23826713455113851746778791436, 9.93427733099468504892432733418, 10.82964401910017115690445683307, 11.17088186550892381632134634000, 11.882197185380474371431089915543, 12.557613584908068187136305409215, 13.32038571199402033734595138686, 13.61209626600573853551357912150, 15.28198913198817552255133770636, 15.82789894773357143393305603016, 16.43124560750607303060704890878, 17.08760568422981485526400363813, 17.31490771863563969626891676137, 18.0607469726238197079000687686

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