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Properties

Label 1-2009-2009.9-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.290 - 0.956i$
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  + (−0.0747 − 0.997i)2-s + (−0.294 + 0.955i)3-s + (−0.988 + 0.149i)4-s + (0.733 − 0.680i)5-s + (0.974 + 0.222i)6-s + (0.222 + 0.974i)8-s + (−0.826 − 0.563i)9-s + (−0.733 − 0.680i)10-s + (0.563 + 0.826i)11-s + (0.149 − 0.988i)12-s + (0.433 − 0.900i)13-s + (0.433 + 0.900i)15-s + (0.955 − 0.294i)16-s + (0.930 − 0.365i)17-s + (−0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.294 + 0.955i)3-s + (−0.988 + 0.149i)4-s + (0.733 − 0.680i)5-s + (0.974 + 0.222i)6-s + (0.222 + 0.974i)8-s + (−0.826 − 0.563i)9-s + (−0.733 − 0.680i)10-s + (0.563 + 0.826i)11-s + (0.149 − 0.988i)12-s + (0.433 − 0.900i)13-s + (0.433 + 0.900i)15-s + (0.955 − 0.294i)16-s + (0.930 − 0.365i)17-s + (−0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.290 - 0.956i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.290 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.162147234 - 0.8618140012i\)
\(L(\frac12)\) \(\approx\) \(1.162147234 - 0.8618140012i\)
\(L(1)\) \(\approx\) \(0.9509755837 - 0.3465279608i\)
\(L(1)\) \(\approx\) \(0.9509755837 - 0.3465279608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
3 \( 1 + (-0.294 + 0.955i)T \)
5 \( 1 + (0.733 - 0.680i)T \)
11 \( 1 + (0.563 + 0.826i)T \)
13 \( 1 + (0.433 - 0.900i)T \)
17 \( 1 + (0.930 - 0.365i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.781 + 0.623i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.997 - 0.0747i)T \)
53 \( 1 + (-0.149 - 0.988i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.781 + 0.623i)T \)
73 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.563 + 0.826i)T \)
97 \( 1 - iT \)
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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

−19.57303507083279629407070032308, −19.00102112714614472266060167710, −18.667707656906245539758377974772, −17.79386181239999045698536916835, −17.04810136222719290240638947115, −16.85752137551903400073477574325, −15.768351248008302318967276287820, −14.80693954579066126578446552360, −14.14194990236067954825994679170, −13.67009854278743534138721558156, −13.08276009293785647541592478949, −12.07197135887503186913506155960, −11.197985625548962689346815548515, −10.45656665378584970546033661772, −9.36087610493176932932166851365, −8.78980774855185500665442482919, −7.85539847697739808114314520323, −7.14183408408149088622548218051, −6.28754655942301058038907638134, −6.09673131643156727363247103841, −5.21370774221639253482774155603, −4.006243807164463006982642338346, −3.027487571400912734322067355739, −1.81710197035701489345628528374, −0.97243766781235433874327328267, 0.67543398146244936166831604716, 1.637283531159535978609085967839, 2.68220925047380951608642043506, 3.60194011265904397901216175665, 4.36145858153022875215718577161, 5.175179572557576163218730158641, 5.63673195801957243593901761373, 6.81234649375925738645956659790, 8.33524034645417119772836601606, 8.75724668012943640400136816008, 9.55707276951442004176059122997, 10.339548484179450252390166616040, 10.513683883870489303991437324785, 11.679436977756626807492752054679, 12.5717069507346131736962931106, 12.67656125655504884493973862821, 14.13009152022952559170548358952, 14.35533610681998499396392803870, 15.42338851001462311615431056068, 16.38339050308038060140053777653, 17.04583892270546017546387846835, 17.57543262038129676414030085467, 18.226175489402082136774933631458, 19.23167705047250564412940258136, 20.18601935846473963158381012892

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