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Properties

Label 1-3009-3009.155-r0-0-0
Degree $1$
Conductor $3009$
Sign $-0.768 + 0.640i$
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  + (−0.319 − 0.947i)2-s + (−0.796 + 0.605i)4-s + (−0.918 − 0.395i)5-s + (−0.344 − 0.938i)7-s + (0.827 + 0.561i)8-s + (−0.0811 + 0.996i)10-s + (−0.492 + 0.870i)11-s + (0.468 + 0.883i)13-s + (−0.779 + 0.626i)14-s + (0.267 − 0.963i)16-s + (−0.214 + 0.976i)19-s + (0.970 − 0.241i)20-s + (0.982 + 0.188i)22-s + (0.583 − 0.812i)23-s + (0.687 + 0.725i)25-s + (0.687 − 0.725i)26-s + ⋯
L(s)  = 1  + (−0.319 − 0.947i)2-s + (−0.796 + 0.605i)4-s + (−0.918 − 0.395i)5-s + (−0.344 − 0.938i)7-s + (0.827 + 0.561i)8-s + (−0.0811 + 0.996i)10-s + (−0.492 + 0.870i)11-s + (0.468 + 0.883i)13-s + (−0.779 + 0.626i)14-s + (0.267 − 0.963i)16-s + (−0.214 + 0.976i)19-s + (0.970 − 0.241i)20-s + (0.982 + 0.188i)22-s + (0.583 − 0.812i)23-s + (0.687 + 0.725i)25-s + (0.687 − 0.725i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3009\)    =    \(3 \cdot 17 \cdot 59\)
Sign: $-0.768 + 0.640i$
Analytic conductor: \(13.9737\)
Root analytic conductor: \(13.9737\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3009} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3009,\ (0:\ ),\ -0.768 + 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03634176260 - 0.1003994825i\)
\(L(\frac12)\) \(\approx\) \(-0.03634176260 - 0.1003994825i\)
\(L(1)\) \(\approx\) \(0.5325430699 - 0.2481493207i\)
\(L(1)\) \(\approx\) \(0.5325430699 - 0.2481493207i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.319 - 0.947i)T \)
5 \( 1 + (-0.918 - 0.395i)T \)
7 \( 1 + (-0.344 - 0.938i)T \)
11 \( 1 + (-0.492 + 0.870i)T \)
13 \( 1 + (0.468 + 0.883i)T \)
19 \( 1 + (-0.214 + 0.976i)T \)
23 \( 1 + (0.583 - 0.812i)T \)
29 \( 1 + (-0.895 + 0.444i)T \)
31 \( 1 + (-0.538 + 0.842i)T \)
37 \( 1 + (0.982 + 0.188i)T \)
41 \( 1 + (-0.812 + 0.583i)T \)
43 \( 1 + (0.963 + 0.267i)T \)
47 \( 1 + (0.370 - 0.928i)T \)
53 \( 1 + (-0.762 - 0.647i)T \)
61 \( 1 + (-0.895 - 0.444i)T \)
67 \( 1 + (0.561 - 0.827i)T \)
71 \( 1 + (-0.395 - 0.918i)T \)
73 \( 1 + (-0.779 + 0.626i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (0.998 + 0.0541i)T \)
89 \( 1 + (0.947 + 0.319i)T \)
97 \( 1 + (-0.626 + 0.779i)T \)
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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

−19.08034294074230318148237668963, −18.84360319020494356217168609544, −18.14554813626933632753743126610, −17.40234386941576616965462318846, −16.52589904593325711146897700450, −15.80535632823381306093150883806, −15.38827622296619714879833884886, −15.00360471246858134256542399326, −14.01720397109718095372721539416, −13.13025387288840114004870667292, −12.741747716261595857849240728305, −11.45719806822131454631290278780, −11.04180334539509567663750070961, −10.184255459980112586643503070486, −9.10294854428885798124935710583, −8.79080473336039387536623210261, −7.76391920261742670289351544848, −7.514173962211456069348528139806, −6.37493305414630187650225116618, −5.80293456355555463378458406639, −5.15454408764738113240924838029, −4.09680258217923445112252537150, −3.28023540523618830664099287378, −2.48325497374662343183719092899, −0.9313553257533352732973439756, 0.0489461515547145448360937443, 1.20247092606387305719143339941, 1.93803958774585814699838351064, 3.16386387770420717214831157126, 3.78632144037508208474369535811, 4.45637682820372390691427610740, 5.03335849447466889842504386851, 6.474063924310091437998171148441, 7.36626573737473897755651667149, 7.84724148797155706845984419907, 8.7327795616591271848549117330, 9.383494104999845259770531356640, 10.22822071346004768781482064915, 10.83654068280021826327166378052, 11.45272093571872514902375861117, 12.36425563725353881227684624377, 12.75261761379985724825872541688, 13.453090856089428331717169680014, 14.34853935384585655876594562539, 15.03293780339903645376776069650, 16.271698613198799609591056619418, 16.49049525358711335690504860109, 17.213584695350955131433261876341, 18.20033002383626336176113007013, 18.7556556903865620879502286953

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