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Properties

Label 1-3009-3009.644-r0-0-0
Degree $1$
Conductor $3009$
Sign $0.292 + 0.956i$
Analytic cond. $13.9737$
Root an. cond. $13.9737$
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  + (−0.605 + 0.796i)2-s + (−0.267 − 0.963i)4-s + (0.999 − 0.0270i)5-s + (0.996 − 0.0811i)7-s + (0.928 + 0.370i)8-s + (−0.583 + 0.812i)10-s + (0.241 + 0.970i)11-s + (−0.561 − 0.827i)13-s + (−0.538 + 0.842i)14-s + (−0.856 + 0.515i)16-s + (0.419 + 0.907i)19-s + (−0.293 − 0.955i)20-s + (−0.918 − 0.395i)22-s + (0.444 + 0.895i)23-s + (0.998 − 0.0541i)25-s + (0.998 + 0.0541i)26-s + ⋯
L(s)  = 1  + (−0.605 + 0.796i)2-s + (−0.267 − 0.963i)4-s + (0.999 − 0.0270i)5-s + (0.996 − 0.0811i)7-s + (0.928 + 0.370i)8-s + (−0.583 + 0.812i)10-s + (0.241 + 0.970i)11-s + (−0.561 − 0.827i)13-s + (−0.538 + 0.842i)14-s + (−0.856 + 0.515i)16-s + (0.419 + 0.907i)19-s + (−0.293 − 0.955i)20-s + (−0.918 − 0.395i)22-s + (0.444 + 0.895i)23-s + (0.998 − 0.0541i)25-s + (0.998 + 0.0541i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3009\)    =    \(3 \cdot 17 \cdot 59\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(13.9737\)
Root analytic conductor: \(13.9737\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3009} (644, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3009,\ (0:\ ),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454332082 + 1.075825670i\)
\(L(\frac12)\) \(\approx\) \(1.454332082 + 1.075825670i\)
\(L(1)\) \(\approx\) \(1.022954199 + 0.4129442513i\)
\(L(1)\) \(\approx\) \(1.022954199 + 0.4129442513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.605 + 0.796i)T \)
5 \( 1 + (0.999 - 0.0270i)T \)
7 \( 1 + (0.996 - 0.0811i)T \)
11 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (-0.561 - 0.827i)T \)
19 \( 1 + (0.419 + 0.907i)T \)
23 \( 1 + (0.444 + 0.895i)T \)
29 \( 1 + (-0.135 + 0.990i)T \)
31 \( 1 + (0.938 - 0.344i)T \)
37 \( 1 + (-0.918 - 0.395i)T \)
41 \( 1 + (0.895 + 0.444i)T \)
43 \( 1 + (-0.515 - 0.856i)T \)
47 \( 1 + (0.725 - 0.687i)T \)
53 \( 1 + (0.986 + 0.161i)T \)
61 \( 1 + (-0.135 - 0.990i)T \)
67 \( 1 + (0.370 - 0.928i)T \)
71 \( 1 + (-0.0270 + 0.999i)T \)
73 \( 1 + (-0.538 + 0.842i)T \)
79 \( 1 + (-0.955 + 0.293i)T \)
83 \( 1 + (-0.108 - 0.994i)T \)
89 \( 1 + (-0.796 + 0.605i)T \)
97 \( 1 + (-0.842 + 0.538i)T \)
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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

−19.035646604986721029675708587757, −18.122667508655319565058642220525, −17.688623507101707255604134267027, −16.96295594846131444423027763885, −16.56514957387685443287362010484, −15.48489504942663370200961396792, −14.4177634662246119821020180958, −13.88613812479451700540990291317, −13.36816262354256493076447355508, −12.39494744188811585478479685621, −11.641243584383411332773167922749, −11.14193571657315031491581432195, −10.388792859579133086800851042478, −9.683507206854190755096950434442, −8.83296174392464846665421276960, −8.57218182195460308464073145007, −7.47164806931396303050790838979, −6.74431530282136964365013078924, −5.77471885609610162629901471515, −4.79977649578189631310417987742, −4.26534908066134105879137920934, −2.94045412612435955746902014454, −2.44265843778559443894009908522, −1.54924018704703386277140670799, −0.793017980371210640478273958521, 1.060549213495073794688635328440, 1.69804829654964149564083619197, 2.49658755189115690853161225810, 3.9174328660163705010496862185, 5.11080393811718237677241853470, 5.21727744705987184571028471261, 6.13633937195110180632687988814, 7.15692228473885980738649474187, 7.5276573246962855009589089631, 8.46305130300781840793629122825, 9.12704273065978744177615398893, 10.02209859510979177034014290646, 10.26021053173073907951552388945, 11.21012759146268499933987081905, 12.18293864582843891808124141765, 13.00337183461323303002981108482, 13.93193370675341439953866259614, 14.34836598031775549712076524372, 15.057237679129337745791515473, 15.59140667909570924132374455999, 16.71180693966593051879803338365, 17.25213177720303250356375648073, 17.661205903098754769803636711253, 18.22696995538441062766425844102, 18.9151431276897911987312644785

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