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Properties

Label 1-4009-4009.1826-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.999 - 0.0340i$
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  + (−0.661 + 0.749i)2-s + (0.980 + 0.198i)3-s + (−0.124 − 0.992i)4-s + (−0.0249 − 0.999i)5-s + (−0.797 + 0.603i)6-s + (−0.623 + 0.781i)7-s + (0.826 + 0.563i)8-s + (0.921 + 0.388i)9-s + (0.766 + 0.642i)10-s + (0.365 − 0.930i)11-s + (0.0747 − 0.997i)12-s + (0.583 − 0.811i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + (−0.969 + 0.246i)16-s + (0.583 + 0.811i)17-s + ⋯
L(s)  = 1  + (−0.661 + 0.749i)2-s + (0.980 + 0.198i)3-s + (−0.124 − 0.992i)4-s + (−0.0249 − 0.999i)5-s + (−0.797 + 0.603i)6-s + (−0.623 + 0.781i)7-s + (0.826 + 0.563i)8-s + (0.921 + 0.388i)9-s + (0.766 + 0.642i)10-s + (0.365 − 0.930i)11-s + (0.0747 − 0.997i)12-s + (0.583 − 0.811i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + (−0.969 + 0.246i)16-s + (0.583 + 0.811i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.999 - 0.0340i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1826, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.999 - 0.0340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.751928424 + 0.02980662714i\)
\(L(\frac12)\) \(\approx\) \(1.751928424 + 0.02980662714i\)
\(L(1)\) \(\approx\) \(1.075308261 + 0.1651685591i\)
\(L(1)\) \(\approx\) \(1.075308261 + 0.1651685591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.661 + 0.749i)T \)
3 \( 1 + (0.980 + 0.198i)T \)
5 \( 1 + (-0.0249 - 0.999i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (0.583 - 0.811i)T \)
17 \( 1 + (0.583 + 0.811i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.583 + 0.811i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.988 + 0.149i)T \)
41 \( 1 + (0.995 - 0.0995i)T \)
43 \( 1 + (-0.583 - 0.811i)T \)
47 \( 1 + (0.456 - 0.889i)T \)
53 \( 1 + (0.998 - 0.0498i)T \)
59 \( 1 + (0.583 + 0.811i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.318 + 0.947i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.542 - 0.840i)T \)
79 \( 1 + (-0.270 + 0.962i)T \)
83 \( 1 + T \)
89 \( 1 + (0.878 - 0.478i)T \)
97 \( 1 + (0.542 - 0.840i)T \)
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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

−18.59706757968474290826998920247, −18.07199273475621803433907226619, −17.40305840728808375221306637412, −16.37858671025246309947134478572, −15.95922761000357008271955927378, −14.91210598708669331797453340203, −14.30151612317225130853704167629, −13.585488708672969991709039890043, −13.14939373532288917413692796276, −12.22774552958089001535543932593, −11.51798095965169654025782637992, −10.83828516616364731456594911448, −9.808578038652745823059239548866, −9.72282253395448432313774319578, −9.05590401395190828752142052210, −7.74561631823651720063816398990, −7.59008951397880517931309346384, −6.87399155546504146782823693124, −6.15226369109193153728421177558, −4.41007690529473771632701613337, −3.861465872284653786898961368081, −3.356952296712080453377245061706, −2.440213582916128344553434892333, −1.88877329955807680287193692551, −0.90058470471809823449960541195, 0.669321540330676772520215127782, 1.53711897668994280092595740044, 2.413583332558541702212420924466, 3.53165955740766482344179529410, 4.09384182452086135751943494436, 5.37713298515182707952851225169, 5.6758385822169802686022779182, 6.53353989193825064479356483107, 7.573234074656003279948903636, 8.246098144576929186231532381166, 8.81871444381808791692692493482, 9.001788150896092827028377101449, 10.009840221586575724868607228031, 10.471453106822726155398661607087, 11.5593384897371863553638854173, 12.6122360869637505261663104128, 13.08377757667788333104948860360, 13.799633576895958310022047731322, 14.6473142944463449767505973290, 15.11684179542194714824956186882, 15.96173218239063862994601767453, 16.32161831083598089574854005701, 16.79539939044405717621961178084, 17.925362104229132444534399090505, 18.51240954779770009984897740716

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