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Properties

Label 1-4009-4009.1078-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.908 + 0.417i$
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  + (−0.997 + 0.0697i)2-s + (−0.615 + 0.788i)3-s + (0.990 − 0.139i)4-s + (−0.882 + 0.469i)5-s + (0.559 − 0.829i)6-s + (0.104 + 0.994i)7-s + (−0.978 + 0.207i)8-s + (−0.241 − 0.970i)9-s + (0.848 − 0.529i)10-s + (0.669 − 0.743i)11-s + (−0.5 + 0.866i)12-s + (−0.559 + 0.829i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + (0.961 − 0.275i)16-s + (0.241 − 0.970i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0697i)2-s + (−0.615 + 0.788i)3-s + (0.990 − 0.139i)4-s + (−0.882 + 0.469i)5-s + (0.559 − 0.829i)6-s + (0.104 + 0.994i)7-s + (−0.978 + 0.207i)8-s + (−0.241 − 0.970i)9-s + (0.848 − 0.529i)10-s + (0.669 − 0.743i)11-s + (−0.5 + 0.866i)12-s + (−0.559 + 0.829i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + (0.961 − 0.275i)16-s + (0.241 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.908 + 0.417i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1078, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.908 + 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4631468607 + 0.1014008220i\)
\(L(\frac12)\) \(\approx\) \(0.4631468607 + 0.1014008220i\)
\(L(1)\) \(\approx\) \(0.4409866152 + 0.1700136134i\)
\(L(1)\) \(\approx\) \(0.4409866152 + 0.1700136134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.997 + 0.0697i)T \)
3 \( 1 + (-0.615 + 0.788i)T \)
5 \( 1 + (-0.882 + 0.469i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (-0.559 + 0.829i)T \)
17 \( 1 + (0.241 - 0.970i)T \)
23 \( 1 + (0.719 - 0.694i)T \)
29 \( 1 + (-0.997 - 0.0697i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.0348 - 0.999i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.997 - 0.0697i)T \)
53 \( 1 + (0.882 + 0.469i)T \)
59 \( 1 + (-0.848 - 0.529i)T \)
61 \( 1 + (0.882 + 0.469i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.990 - 0.139i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.0348 + 0.999i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (0.559 - 0.829i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−18.41701722525103846853455330456, −17.46561369987308681043508996116, −17.28356539189793432309621363935, −16.68145503313783274825780224165, −16.06309808684394515756736956732, −14.991778944328215038654593663890, −14.713749914149747077965647932817, −13.27769423088945179475873589537, −12.77137235804234079886290971371, −12.18446830588175541922835604833, −11.42730471106997300899412909086, −10.96139272224755723171661113033, −10.21415883058419362815908721363, −9.44549246683829562242506129453, −8.50893821217616866479269286293, −7.75929012041606645768505348114, −7.389858749460721919711817244654, −6.88639044754675760862013761609, −5.88105770795266025080498730213, −5.07906625307438033633205808821, −4.04682697947484002018605283616, −3.33155857532799518625897194253, −2.043874742324786417651431935935, −1.358323114716794885143540947786, −0.61318706769137292912072289999, 0.357624186982546655885733482293, 1.53274393626326761579888853839, 2.78292970455036644120016036744, 3.23196473319198621873829120841, 4.27661879956693364078885732618, 5.16564390597289686992033109664, 5.91758822040549580246210583025, 6.76928643770728614573572402296, 7.17446561604674942404667776272, 8.34623239792615911943238397043, 8.88925076272402198678973624609, 9.42596279366195555523787062971, 10.20234829871665536574453955813, 11.11215949947038495259655138958, 11.45884286809751225809754361777, 11.9303137800845668629506707123, 12.58500028477410131798173582104, 14.243234054472016176313265118943, 14.71689946866372769735989467373, 15.2810723322834545829775676444, 16.094438139773572813843890430765, 16.37595520266414377462340169017, 17.05641074337429828180104469725, 17.886728417124204372615750603007, 18.65702868724995962033173676933

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