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Properties

Label 1-2009-2009.2-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.526 - 0.850i$
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  + (0.999 − 0.0299i)2-s + (−0.680 + 0.733i)3-s + (0.998 − 0.0598i)4-s + (−0.0149 + 0.999i)5-s + (−0.657 + 0.753i)6-s + (0.995 − 0.0896i)8-s + (−0.0747 − 0.997i)9-s + (0.0149 + 0.999i)10-s + (−0.237 − 0.971i)11-s + (−0.635 + 0.772i)12-s + (−0.880 − 0.473i)13-s + (−0.722 − 0.691i)15-s + (0.992 − 0.119i)16-s + (0.894 − 0.447i)17-s + (−0.104 − 0.994i)18-s + (−0.994 − 0.104i)19-s + ⋯
L(s)  = 1  + (0.999 − 0.0299i)2-s + (−0.680 + 0.733i)3-s + (0.998 − 0.0598i)4-s + (−0.0149 + 0.999i)5-s + (−0.657 + 0.753i)6-s + (0.995 − 0.0896i)8-s + (−0.0747 − 0.997i)9-s + (0.0149 + 0.999i)10-s + (−0.237 − 0.971i)11-s + (−0.635 + 0.772i)12-s + (−0.880 − 0.473i)13-s + (−0.722 − 0.691i)15-s + (0.992 − 0.119i)16-s + (0.894 − 0.447i)17-s + (−0.104 − 0.994i)18-s + (−0.994 − 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.526 - 0.850i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.526 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429183181 - 0.7962221304i\)
\(L(\frac12)\) \(\approx\) \(1.429183181 - 0.7962221304i\)
\(L(1)\) \(\approx\) \(1.369401599 + 0.1129119966i\)
\(L(1)\) \(\approx\) \(1.369401599 + 0.1129119966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.999 - 0.0299i)T \)
3 \( 1 + (-0.680 + 0.733i)T \)
5 \( 1 + (-0.0149 + 0.999i)T \)
11 \( 1 + (-0.237 - 0.971i)T \)
13 \( 1 + (-0.880 - 0.473i)T \)
17 \( 1 + (0.894 - 0.447i)T \)
19 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (0.887 - 0.460i)T \)
29 \( 1 + (-0.351 - 0.936i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.772 - 0.635i)T \)
43 \( 1 + (-0.753 - 0.657i)T \)
47 \( 1 + (-0.0299 - 0.999i)T \)
53 \( 1 + (-0.0598 - 0.998i)T \)
59 \( 1 + (-0.946 + 0.323i)T \)
61 \( 1 + (0.842 - 0.538i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (0.351 - 0.936i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.850 - 0.525i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−20.069264380086383962544654701411, −19.48572605245879329677795445406, −18.765861676022081796861403375817, −17.6032645811419587635339945886, −16.89836112953244825307562095851, −16.66388289015008675253259513613, −15.657204845862462191063472906161, −14.82946908567423277946855541362, −14.14011637360314033794735610647, −13.09337670113826681287755516067, −12.691828743269840881391191068855, −12.26849782944147820658267614915, −11.52502078323847471725667869999, −10.64415631395491280018102288895, −9.8263665122020712126533949147, −8.67425138797655470294342012004, −7.625739902529572383671701872306, −7.21081535602573722333932225268, −6.28430312379826911026858958929, −5.40794893076521382220966138330, −4.89777939957911187949112526788, −4.23652234458724756815005093396, −2.97228987908656480605482883856, −1.80150633246996240842942551072, −1.43637173211411073650072196532, 0.39350041147099782130198352698, 2.084127280319604517349648042242, 3.06246316196173599814054445045, 3.5414722218951308516091182771, 4.502275977335351721047201560, 5.442922824750224967067893506720, 5.8392488912093396782179993422, 6.80872280234318924713868857078, 7.386841141749764633641153591629, 8.53885789863729414830147447041, 9.84039465372309950342596641969, 10.38348353132542290666160360385, 11.05035897399342937212267679548, 11.58497667784374447484413326940, 12.40086925258881839526113852364, 13.201164670656943149333739798, 14.16697443820943706935549603507, 14.80509100035854926234756627075, 15.224505702736666045332150939257, 16.06398510559665290171525334505, 16.80587660261666371801263784891, 17.34032280368190065064816897597, 18.523551226684477728909561881, 19.08955047676342036978020200700, 19.98151684640586479679166839435

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