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Properties

Label 1-3009-3009.308-r0-0-0
Degree $1$
Conductor $3009$
Sign $-0.999 + 0.0176i$
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  + (−0.986 − 0.161i)2-s + (0.947 + 0.319i)4-s + (−0.982 − 0.188i)5-s + (−0.842 − 0.538i)7-s + (−0.883 − 0.468i)8-s + (0.938 + 0.344i)10-s + (0.990 + 0.135i)11-s + (−0.856 + 0.515i)13-s + (0.744 + 0.667i)14-s + (0.796 + 0.605i)16-s + (−0.108 − 0.994i)19-s + (−0.870 − 0.492i)20-s + (−0.955 − 0.293i)22-s + (−0.0811 + 0.996i)23-s + (0.928 + 0.370i)25-s + (0.928 − 0.370i)26-s + ⋯
L(s)  = 1  + (−0.986 − 0.161i)2-s + (0.947 + 0.319i)4-s + (−0.982 − 0.188i)5-s + (−0.842 − 0.538i)7-s + (−0.883 − 0.468i)8-s + (0.938 + 0.344i)10-s + (0.990 + 0.135i)11-s + (−0.856 + 0.515i)13-s + (0.744 + 0.667i)14-s + (0.796 + 0.605i)16-s + (−0.108 − 0.994i)19-s + (−0.870 − 0.492i)20-s + (−0.955 − 0.293i)22-s + (−0.0811 + 0.996i)23-s + (0.928 + 0.370i)25-s + (0.928 − 0.370i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3009\)    =    \(3 \cdot 17 \cdot 59\)
Sign: $-0.999 + 0.0176i$
Analytic conductor: \(13.9737\)
Root analytic conductor: \(13.9737\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3009} (308, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3009,\ (0:\ ),\ -0.999 + 0.0176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001450574361 - 0.1643659283i\)
\(L(\frac12)\) \(\approx\) \(0.001450574361 - 0.1643659283i\)
\(L(1)\) \(\approx\) \(0.4732230269 - 0.08872477604i\)
\(L(1)\) \(\approx\) \(0.4732230269 - 0.08872477604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.986 - 0.161i)T \)
5 \( 1 + (-0.982 - 0.188i)T \)
7 \( 1 + (-0.842 - 0.538i)T \)
11 \( 1 + (0.990 + 0.135i)T \)
13 \( 1 + (-0.856 + 0.515i)T \)
19 \( 1 + (-0.108 - 0.994i)T \)
23 \( 1 + (-0.0811 + 0.996i)T \)
29 \( 1 + (-0.812 - 0.583i)T \)
31 \( 1 + (0.779 - 0.626i)T \)
37 \( 1 + (-0.955 - 0.293i)T \)
41 \( 1 + (0.996 - 0.0811i)T \)
43 \( 1 + (-0.605 + 0.796i)T \)
47 \( 1 + (0.561 - 0.827i)T \)
53 \( 1 + (0.419 - 0.907i)T \)
61 \( 1 + (-0.812 + 0.583i)T \)
67 \( 1 + (-0.468 + 0.883i)T \)
71 \( 1 + (-0.188 - 0.982i)T \)
73 \( 1 + (0.744 + 0.667i)T \)
79 \( 1 + (-0.492 + 0.870i)T \)
83 \( 1 + (0.687 - 0.725i)T \)
89 \( 1 + (0.161 + 0.986i)T \)
97 \( 1 + (-0.667 - 0.744i)T \)
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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

−19.28423305041739989951767470252, −18.80745542170626228931843306666, −18.215461208133194137964075659140, −17.140182456819635827731078887562, −16.710258457143485520152787541809, −15.98627588315346265606809909876, −15.38662077868690476288730356580, −14.73923580046127232218254138949, −14.13847014260435116339229938763, −12.6186693003434317788347473407, −12.23221909846327302780705878861, −11.707996445884033946730151458174, −10.652112291003855775244721839348, −10.222495794472182288578372466704, −9.238000789425669148636218445237, −8.76582567526530716787456538868, −7.93788496947552619957905630795, −7.25168571850580762203611836290, −6.5350351917272762992138493873, −5.91959452639440369094329931767, −4.83919822880261104546082603091, −3.68036625052383406490601395033, −3.06016784732069616027678216744, −2.17457421129542902855945622516, −1.002311348838960027120475972341, 0.09516115565649812164159034908, 0.99594180714892591780381247526, 2.09390766468855840414720550670, 3.08484226801606395623132262424, 3.83037375523266478215533037713, 4.49428336473854902938164699506, 5.815827219165811490008112327799, 6.85592237846821601386501754222, 7.12312983874143580065544161859, 7.87009047398532788942606037474, 8.83329434092634711374798563947, 9.41722669724379450059428984426, 9.949635510236523359863772246766, 10.95250814396653793291522749736, 11.602750899711936643711360335333, 12.06756943973788988236808889316, 12.87253152957141310423624154058, 13.71592046244444663937543294704, 14.83446841476170355638672854859, 15.36689540692976132719648469112, 16.08041526020588550867918381296, 16.79687502831832980591251919406, 17.14656908223154941668361303516, 17.964256427752020918613742445751, 19.09293553472610856353445408582

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