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Properties

Label 1-2009-2009.795-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.255 - 0.966i$
Analytic cond. $9.32975$
Root an. cond. $9.32975$
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  + (−0.646 + 0.762i)2-s + (0.955 − 0.294i)3-s + (−0.163 − 0.986i)4-s + (0.420 − 0.907i)5-s + (−0.393 + 0.919i)6-s + (0.858 + 0.512i)8-s + (0.826 − 0.563i)9-s + (0.420 + 0.907i)10-s + (0.791 − 0.611i)11-s + (−0.447 − 0.894i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (−0.946 + 0.323i)16-s + (−0.772 − 0.635i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (−0.646 + 0.762i)2-s + (0.955 − 0.294i)3-s + (−0.163 − 0.986i)4-s + (0.420 − 0.907i)5-s + (−0.393 + 0.919i)6-s + (0.858 + 0.512i)8-s + (0.826 − 0.563i)9-s + (0.420 + 0.907i)10-s + (0.791 − 0.611i)11-s + (−0.447 − 0.894i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (−0.946 + 0.323i)16-s + (−0.772 − 0.635i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.255 - 0.966i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.255 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451058855 - 1.117456552i\)
\(L(\frac12)\) \(\approx\) \(1.451058855 - 1.117456552i\)
\(L(1)\) \(\approx\) \(1.189054797 - 0.2066879654i\)
\(L(1)\) \(\approx\) \(1.189054797 - 0.2066879654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.646 + 0.762i)T \)
3 \( 1 + (0.955 - 0.294i)T \)
5 \( 1 + (0.420 - 0.907i)T \)
11 \( 1 + (0.791 - 0.611i)T \)
13 \( 1 + (0.983 + 0.178i)T \)
17 \( 1 + (-0.772 - 0.635i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.251 - 0.967i)T \)
29 \( 1 + (-0.550 + 0.834i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.447 - 0.894i)T \)
43 \( 1 + (-0.393 + 0.919i)T \)
47 \( 1 + (-0.646 + 0.762i)T \)
53 \( 1 + (-0.163 - 0.986i)T \)
59 \( 1 + (0.992 - 0.119i)T \)
61 \( 1 + (0.712 - 0.701i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.550 - 0.834i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.337 + 0.941i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−20.13578988089228036291399885192, −19.29026619902849633862299260792, −18.82041382026587579683496931843, −18.11436631642645068297913173741, −17.38324820187108989692211528148, −16.63708638569056223776116981351, −15.56255504345973098465196231893, −15.01397045873712246273328675487, −14.15841833458477139867850571594, −13.40365625349144104161131668266, −12.914575407866924416470889951610, −11.72697470488368041059315010595, −11.075234797057827200395156248713, −10.20564798341834616205817962874, −9.83131535398307848366277180285, −8.94434827069303514990868479181, −8.37021356107331468317883311243, −7.40934459452207529878874977236, −6.815350247122900988957696595009, −5.69644790927285078568549383449, −4.17418847354529655224648629823, −3.7345022969269863017186863756, −2.9972402478158297817482178744, −1.8423796716605759292682351707, −1.6460711231364375413953379780, 0.68013134651201829435630613585, 1.48393729644381142105430989869, 2.316634905887934098198785399519, 3.63099952437309938638342821064, 4.54492326996479449221354536311, 5.38563113677338699279665223118, 6.51086998451004181790193810470, 6.83224723136591251480335223047, 7.99414386594695003189333537076, 8.72014639712609528166749439613, 9.03891789624571080811684995155, 9.59041179570916594266821358856, 10.78051931318451595157773924068, 11.49680895864234024746027466689, 12.91462461762065908166558879078, 13.20230099743934726974457018546, 14.20234974663908533319229714410, 14.470916786296266563728920104268, 15.65437836910983166517142244726, 16.116528259419829983467311069729, 16.77759318982313394424056118475, 17.76291375455670907702827108030, 18.19986021269599641772568011853, 19.048035692587073070439458117977, 19.79144730193463887333450865138

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