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Properties

Label 1-1009-1009.654-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.991 - 0.127i$
Analytic cond. $4.68577$
Root an. cond. $4.68577$
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 + (−0.733 + 0.680i)3-s + (−0.969 − 0.246i)4-s + (−0.411 + 0.911i)5-s + (−0.583 − 0.811i)6-s + (0.542 + 0.840i)7-s + (0.365 − 0.930i)8-s + (0.0747 − 0.997i)9-s + (−0.853 − 0.521i)10-s + (0.411 − 0.911i)11-s + (0.878 − 0.478i)12-s + (0.5 − 0.866i)13-s + (−0.900 + 0.433i)14-s + (−0.318 − 0.947i)15-s + (0.878 + 0.478i)16-s + (−0.878 − 0.478i)17-s + ⋯
L(s)  = 1  + (−0.124 + 0.992i)2-s + (−0.733 + 0.680i)3-s + (−0.969 − 0.246i)4-s + (−0.411 + 0.911i)5-s + (−0.583 − 0.811i)6-s + (0.542 + 0.840i)7-s + (0.365 − 0.930i)8-s + (0.0747 − 0.997i)9-s + (−0.853 − 0.521i)10-s + (0.411 − 0.911i)11-s + (0.878 − 0.478i)12-s + (0.5 − 0.866i)13-s + (−0.900 + 0.433i)14-s + (−0.318 − 0.947i)15-s + (0.878 + 0.478i)16-s + (−0.878 − 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.991 - 0.127i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05353430365 + 0.8337190398i\)
\(L(\frac12)\) \(\approx\) \(-0.05353430365 + 0.8337190398i\)
\(L(1)\) \(\approx\) \(0.4588155867 + 0.5735286112i\)
\(L(1)\) \(\approx\) \(0.4588155867 + 0.5735286112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (-0.124 + 0.992i)T \)
3 \( 1 + (-0.733 + 0.680i)T \)
5 \( 1 + (-0.411 + 0.911i)T \)
7 \( 1 + (0.542 + 0.840i)T \)
11 \( 1 + (0.411 - 0.911i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.878 - 0.478i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.988 + 0.149i)T \)
29 \( 1 + (0.456 + 0.889i)T \)
31 \( 1 + (0.124 + 0.992i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (0.797 + 0.603i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (0.542 - 0.840i)T \)
71 \( 1 + (-0.124 + 0.992i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (0.318 + 0.947i)T \)
83 \( 1 + (0.853 - 0.521i)T \)
89 \( 1 + (0.583 + 0.811i)T \)
97 \( 1 + (-0.698 - 0.715i)T \)
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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

−20.96288665713113816941388346644, −20.49268327596596250321348933668, −19.5239832934351424449138901214, −19.20761153801610657202484482843, −17.960285243882095051107263536075, −17.415768574329139878765512537818, −16.958469824130495596398142437149, −15.98323877727458650902790492278, −14.72061535451772349729635171727, −13.48597097773080811110954784513, −13.282249685422625914987040512833, −12.32234006324158592936145303741, −11.481086211876775867883905789666, −11.23153065978539424178003179405, −10.11940491537097356797934728677, −9.128509906301006527921231293106, −8.31954487750793658677825028909, −7.42849361611538362364166354539, −6.572926703279138312020505931294, −5.07982953253923595730159909115, −4.5749666460150046228973758846, −3.85403816872633561851483966417, −2.14166418153815915406124979751, −1.43982891820190904069254780659, −0.53109103040780912755181765162, 1.08586393390895658057709717528, 3.08423180431133959414076283920, 3.78027345804378595259886499568, 5.02611947124817855325390367193, 5.55986568058423507079325780783, 6.454720330261092790482357349124, 7.130959671187741739514134437352, 8.43736731615830647737330977894, 8.81955361201707617750459115494, 10.04691423006633177084142312064, 10.77284397411381979179800296378, 11.51958908119576972589336454913, 12.36964761557409278611948560162, 13.63880342117378857919442269263, 14.48530926790821684694957885432, 15.14992786462562794347315946310, 15.75203672857104146857330510745, 16.31851803444157641472254509899, 17.38092973326051045859159086029, 18.0625667380302966142800014375, 18.506532599455762780759863663057, 19.41844853050678669590425101634, 20.66762042459681983618344353022, 21.71909617369944924172825731753, 22.134159458413108669901583877961

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