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Properties

Label 1-4009-4009.1043-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.965 + 0.262i$
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  + (−0.124 + 0.992i)2-s + (0.921 − 0.388i)3-s + (−0.969 − 0.246i)4-s + (0.456 + 0.889i)5-s + (0.270 + 0.962i)6-s + (−0.733 − 0.680i)7-s + (0.365 − 0.930i)8-s + (0.698 − 0.715i)9-s + (−0.939 + 0.342i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.661 − 0.749i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)15-s + (0.878 + 0.478i)16-s + (−0.318 − 0.947i)17-s + ⋯
L(s)  = 1  + (−0.124 + 0.992i)2-s + (0.921 − 0.388i)3-s + (−0.969 − 0.246i)4-s + (0.456 + 0.889i)5-s + (0.270 + 0.962i)6-s + (−0.733 − 0.680i)7-s + (0.365 − 0.930i)8-s + (0.698 − 0.715i)9-s + (−0.939 + 0.342i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.661 − 0.749i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)15-s + (0.878 + 0.478i)16-s + (−0.318 − 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.965 + 0.262i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.965 + 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1269617843 + 0.9522080041i\)
\(L(\frac12)\) \(\approx\) \(0.1269617843 + 0.9522080041i\)
\(L(1)\) \(\approx\) \(0.9064037372 + 0.4492263182i\)
\(L(1)\) \(\approx\) \(0.9064037372 + 0.4492263182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.124 + 0.992i)T \)
3 \( 1 + (0.921 - 0.388i)T \)
5 \( 1 + (0.456 + 0.889i)T \)
7 \( 1 + (-0.733 - 0.680i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.661 - 0.749i)T \)
17 \( 1 + (-0.318 - 0.947i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.318 + 0.947i)T \)
31 \( 1 + (-0.988 + 0.149i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.661 + 0.749i)T \)
43 \( 1 + (0.980 + 0.198i)T \)
47 \( 1 + (0.995 + 0.0995i)T \)
53 \( 1 + (-0.411 - 0.911i)T \)
59 \( 1 + (-0.318 - 0.947i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.124 - 0.992i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.583 - 0.811i)T \)
79 \( 1 + (0.878 + 0.478i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.998 + 0.0498i)T \)
97 \( 1 + (0.995 - 0.0995i)T \)
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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

−18.64548927069094581373283908751, −17.48265379094881195658093944923, −16.83676506191851156599290153503, −16.19089552674474874093789517897, −15.44483658409535189384292342883, −14.60670129511333149563826357745, −13.86009547916099757584228647782, −13.26220702814640531605774247159, −12.693045314366884054326017938257, −12.27013620531810770485997025071, −11.141960181197246327443841097778, −10.438688660487390426138275315792, −9.7758813019117741437942399942, −9.04755002423567290221166622526, −8.8428517989817946333995012267, −8.10721729925006255805423856653, −7.15370712227237807511229303342, −5.79796911889523378144902720954, −5.309454739942512241820557060086, −4.299441169161055179910072993555, −3.84893820362467812015530810606, −2.73142279026372537333235887708, −2.34749013319936253863838671628, −1.58183977572296146945132672271, −0.2518541805993243914102212308, 1.06775644096091298656467432246, 2.22111716004919501559754615362, 3.0977400874736198481382427420, 3.5671513600448808996913968962, 4.74930836675397535744819780170, 5.41857454418077894283376717254, 6.50510089081290730299718043014, 7.03896221572539059929019203562, 7.42971946929509244113330572164, 8.033045384109784188819281662855, 9.251680918519923968527952754720, 9.5420421338972253704188257065, 10.23502569468709264793940737026, 10.88519485610661820271720844754, 12.34343323343070550981009765944, 13.06235755390522803281940301105, 13.399607953677311285589269456203, 14.13854099660211659580659119907, 14.72189081813080020638333186127, 15.343138451701649232933584976108, 15.82404350702201764200057779397, 16.800608193429355579775291210480, 17.54919717169543788214174563886, 18.10847651659912140210712254809, 18.6067333321090680874438484568

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