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Properties

Label 1-2009-2009.257-r0-0-0
Degree $1$
Conductor $2009$
Sign $-0.393 - 0.919i$
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  + (0.0598 − 0.998i)2-s + (−0.757 + 0.652i)3-s + (−0.992 − 0.119i)4-s + (0.0299 + 0.999i)5-s + (0.605 + 0.795i)6-s + (−0.178 + 0.983i)8-s + (0.149 − 0.988i)9-s + (0.999 + 0.0299i)10-s + (−0.301 − 0.953i)11-s + (0.830 − 0.557i)12-s + (0.979 − 0.200i)13-s + (−0.674 − 0.738i)15-s + (0.971 + 0.237i)16-s + (0.989 + 0.141i)17-s + (−0.978 − 0.207i)18-s + (0.544 − 0.838i)19-s + ⋯
L(s)  = 1  + (0.0598 − 0.998i)2-s + (−0.757 + 0.652i)3-s + (−0.992 − 0.119i)4-s + (0.0299 + 0.999i)5-s + (0.605 + 0.795i)6-s + (−0.178 + 0.983i)8-s + (0.149 − 0.988i)9-s + (0.999 + 0.0299i)10-s + (−0.301 − 0.953i)11-s + (0.830 − 0.557i)12-s + (0.979 − 0.200i)13-s + (−0.674 − 0.738i)15-s + (0.971 + 0.237i)16-s + (0.989 + 0.141i)17-s + (−0.978 − 0.207i)18-s + (0.544 − 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ -0.393 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4658304860 - 0.7058739258i\)
\(L(\frac12)\) \(\approx\) \(0.4658304860 - 0.7058739258i\)
\(L(1)\) \(\approx\) \(0.7289530918 - 0.2511347615i\)
\(L(1)\) \(\approx\) \(0.7289530918 - 0.2511347615i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.0598 - 0.998i)T \)
3 \( 1 + (-0.757 + 0.652i)T \)
5 \( 1 + (0.0299 + 0.999i)T \)
11 \( 1 + (-0.301 - 0.953i)T \)
13 \( 1 + (0.979 - 0.200i)T \)
17 \( 1 + (0.989 + 0.141i)T \)
19 \( 1 + (0.544 - 0.838i)T \)
23 \( 1 + (-0.575 - 0.817i)T \)
29 \( 1 + (0.0672 + 0.997i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.193 - 0.981i)T \)
43 \( 1 + (-0.990 - 0.134i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (-0.617 - 0.786i)T \)
59 \( 1 + (-0.791 - 0.611i)T \)
61 \( 1 + (-0.907 + 0.420i)T \)
67 \( 1 + (-0.777 - 0.629i)T \)
71 \( 1 + (0.997 + 0.0672i)T \)
73 \( 1 + (-0.930 + 0.365i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.948 - 0.316i)T \)
97 \( 1 + (-0.156 + 0.987i)T \)
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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

−20.1574880232987443942887803004, −19.1504498817385308001419806593, −18.43068504441348564489479977068, −17.88458233649866161577897646704, −17.12415888623095576869414925807, −16.62607317996927076611608129793, −15.919167929588077587812618759374, −15.36931781662334844875974907820, −14.153456630085329549922743689619, −13.50514576576599961513223751696, −12.99981636881708434102020208786, −12.027808999991991039685431887734, −11.81575205536147546320116403428, −10.20668488851744600414353256791, −9.78272738681416564034837407515, −8.68108874764688500845801758572, −7.84025996767311370314670346544, −7.56330773054904389288419741742, −6.31369541156820517787620416176, −5.896840442627901495084826872283, −5.021005885598527494292973529493, −4.490119272643500433054507534554, −3.39241783907748188584822881436, −1.67929045551108061281653359737, −1.06185568104071456141014844241, 0.388973288241391505759505240639, 1.4640264800130370457465389599, 2.91882251647999369942133879542, 3.28956800732868803790398983309, 4.13503528144145477071841720105, 5.14457724468689194994445174475, 5.881078758502646912963652888935, 6.53066240307128578081490425760, 7.84532056998699954960899656697, 8.68608008685999715290944675719, 9.63075487753618800348389610093, 10.27647050075917132683255417130, 10.94092670444066755819123956239, 11.286746461771563390797150682095, 12.094882206608457502166815341575, 12.96546287838864484307871634229, 13.88121581026107466253733896631, 14.41425517177279028422900739537, 15.3382545302318098774742567023, 16.10287832486551950855927441498, 16.86845182847996989472600366904, 17.90628054429742317025897014660, 18.23982041715318488723714456212, 18.87863961278793752567834781162, 19.733113777885819103407937390353

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