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Properties

Label 1-2009-2009.81-r0-0-0
Degree $1$
Conductor $2009$
Sign $-0.972 + 0.232i$
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  + (−0.988 + 0.149i)2-s + (−0.826 − 0.563i)3-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0747 + 0.997i)10-s + (−0.365 + 0.930i)11-s + (−0.955 − 0.294i)12-s + (−0.623 − 0.781i)13-s + (−0.623 + 0.781i)15-s + (0.826 − 0.563i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (−0.826 − 0.563i)3-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0747 + 0.997i)10-s + (−0.365 + 0.930i)11-s + (−0.955 − 0.294i)12-s + (−0.623 − 0.781i)13-s + (−0.623 + 0.781i)15-s + (0.826 − 0.563i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $-0.972 + 0.232i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ -0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03919588557 - 0.3319240128i\)
\(L(\frac12)\) \(\approx\) \(-0.03919588557 - 0.3319240128i\)
\(L(1)\) \(\approx\) \(0.4414708614 - 0.1962049077i\)
\(L(1)\) \(\approx\) \(0.4414708614 - 0.1962049077i\)

Euler product

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

−20.30572846880814987519934770324, −19.24308623609073566568010396164, −18.86398560023033622483490205654, −18.11658249888451579606769986700, −17.49673841071904092033416613090, −16.714865381497762040957868847313, −16.19413444336616828193073469905, −15.484748327001876403026305100566, −14.65778699268283909180107523303, −13.968477785482739482032256062, −12.60123910183653862064315364923, −11.86530151088227037453869945910, −11.2612900221940554439754493773, −10.69439655204028415150489067320, −9.78604452735073766942573551432, −9.66312628078252158141041825836, −8.290599403479548875572066016, −7.62325430139102559469326648712, −6.6935867511569787458199425014, −6.05768582782664284997930731062, −5.39792832899597853588381513943, −3.88758060600810660489632195903, −3.35749141483256495743690805245, −2.27379627149221190483588487273, −1.20830420014868891354790838391, 0.217727216545926241404521005804, 1.03110473459182468487716800192, 1.98416276027441456036702274397, 2.81610379942246526617275488261, 4.528337090107756699821979904, 5.257664369428536532390312486505, 5.821231494133335382653147227718, 6.9370990538276695304075182669, 7.543048502516678756069711434551, 8.10337938950994845984370355813, 9.138341089963862550794667607, 9.910814688308323736305439291595, 10.43643406949986441600951247311, 11.50183294809331759518208245036, 12.115228070374641228947603755764, 12.64903518581842656288537391837, 13.44276470290065262456757797374, 14.61506818820530951647700791432, 15.513358143918830495545860166901, 16.22147684402391102392643311379, 16.74215856907568677768542592349, 17.474509539391485521791505842191, 17.99215289464969392053735283749, 18.547728410225446671107308926965, 19.515812832046800057038365000607

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