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Properties

Label 1-4009-4009.1013-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.896 - 0.442i$
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  + (0.591 + 0.806i)2-s + (0.868 + 0.495i)3-s + (−0.299 + 0.954i)4-s + (−0.985 + 0.168i)5-s + (0.114 + 0.993i)6-s + (−0.280 + 0.959i)7-s + (−0.946 + 0.323i)8-s + (0.508 + 0.860i)9-s + (−0.719 − 0.694i)10-s + (−0.999 − 0.0299i)11-s + (−0.733 + 0.680i)12-s + (−0.705 + 0.708i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)15-s + (−0.820 − 0.571i)16-s + (−0.356 − 0.934i)17-s + ⋯
L(s)  = 1  + (0.591 + 0.806i)2-s + (0.868 + 0.495i)3-s + (−0.299 + 0.954i)4-s + (−0.985 + 0.168i)5-s + (0.114 + 0.993i)6-s + (−0.280 + 0.959i)7-s + (−0.946 + 0.323i)8-s + (0.508 + 0.860i)9-s + (−0.719 − 0.694i)10-s + (−0.999 − 0.0299i)11-s + (−0.733 + 0.680i)12-s + (−0.705 + 0.708i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)15-s + (−0.820 − 0.571i)16-s + (−0.356 − 0.934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.896 - 0.442i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2662775239 + 0.06218425245i\)
\(L(\frac12)\) \(\approx\) \(-0.2662775239 + 0.06218425245i\)
\(L(1)\) \(\approx\) \(0.6369387940 + 0.8604424295i\)
\(L(1)\) \(\approx\) \(0.6369387940 + 0.8604424295i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.591 + 0.806i)T \)
3 \( 1 + (0.868 + 0.495i)T \)
5 \( 1 + (-0.985 + 0.168i)T \)
7 \( 1 + (-0.280 + 0.959i)T \)
11 \( 1 + (-0.999 - 0.0299i)T \)
13 \( 1 + (-0.705 + 0.708i)T \)
17 \( 1 + (-0.356 - 0.934i)T \)
23 \( 1 + (-0.615 + 0.788i)T \)
29 \( 1 + (-0.261 - 0.965i)T \)
31 \( 1 + (-0.733 + 0.680i)T \)
37 \( 1 + (0.473 - 0.880i)T \)
41 \( 1 + (-0.892 + 0.451i)T \)
43 \( 1 + (0.542 + 0.840i)T \)
47 \( 1 + (-0.183 + 0.983i)T \)
53 \( 1 + (0.383 - 0.923i)T \)
59 \( 1 + (0.837 + 0.546i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (-0.583 - 0.811i)T \)
71 \( 1 + (0.559 - 0.829i)T \)
73 \( 1 + (-0.0249 + 0.999i)T \)
79 \( 1 + (0.999 - 0.0199i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.534 - 0.845i)T \)
97 \( 1 + (-0.429 + 0.903i)T \)
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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

−18.24568338346532902063797959943, −17.28983775840380500053886659608, −16.320866429415610993300833151981, −15.48880713247221174908868072208, −14.95912495145176335287749368921, −14.44488160969901317452366724708, −13.482607898871014740859315007294, −13.04117494664439740327001200814, −12.53463238768360021127634707157, −11.91330948384810560714525301897, −10.85481151833620937861977833692, −10.38755237077500105786702611283, −9.74083743643025960867476774193, −8.66542643545078999714632516725, −8.14720639959422115658044505256, −7.32572980372927167493621180401, −6.77117254431263154800588300428, −5.65726090069218974230596671897, −4.7401310608912080456447585188, −3.98319615511875954125618617054, −3.50712342938954891555913726132, −2.71650262960388598167802402091, −1.96454047793354602934728406883, −0.88132409295180778925192427858, −0.06207526911821737748974688048, 2.20258415633850248940516887983, 2.7011472784137671204718779803, 3.45424735390311627902860293880, 4.18759296416417915653039433728, 4.93183467562987294584768101299, 5.465828597975969402526512214206, 6.603495691811820623719558371210, 7.370275772939659783728263222812, 7.88080455424550103829356845522, 8.49535832515016734750032780495, 9.3108160612240699999720092509, 9.76517737145086457636911343691, 11.06561132235158472346493125102, 11.672259989097563290025338517063, 12.43606817872691246973777648439, 13.083794347991350026539348908929, 13.82046053422496458686553693205, 14.58163032175479122999079883056, 15.04388431429996188581306869615, 15.73014658905541505034932820868, 16.05344198112770276402596693299, 16.57277277194761268691737935670, 17.87506398027566368617714627954, 18.38005447783867278621553194262, 19.18054526249547706442413851943

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