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Properties

Label 1-209-209.59-r1-0-0
Degree $1$
Conductor $209$
Sign $-0.530 + 0.847i$
Analytic cond. $22.4601$
Root an. cond. $22.4601$
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  + (−0.0348 + 0.999i)2-s + (−0.438 + 0.898i)3-s + (−0.997 − 0.0697i)4-s + (−0.374 − 0.927i)5-s + (−0.882 − 0.469i)6-s + (−0.104 − 0.994i)7-s + (0.104 − 0.994i)8-s + (−0.615 − 0.788i)9-s + (0.939 − 0.342i)10-s + (0.5 − 0.866i)12-s + (−0.990 + 0.139i)13-s + (0.997 − 0.0697i)14-s + (0.997 + 0.0697i)15-s + (0.990 + 0.139i)16-s + (−0.615 + 0.788i)17-s + (0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (−0.0348 + 0.999i)2-s + (−0.438 + 0.898i)3-s + (−0.997 − 0.0697i)4-s + (−0.374 − 0.927i)5-s + (−0.882 − 0.469i)6-s + (−0.104 − 0.994i)7-s + (0.104 − 0.994i)8-s + (−0.615 − 0.788i)9-s + (0.939 − 0.342i)10-s + (0.5 − 0.866i)12-s + (−0.990 + 0.139i)13-s + (0.997 − 0.0697i)14-s + (0.997 + 0.0697i)15-s + (0.990 + 0.139i)16-s + (−0.615 + 0.788i)17-s + (0.809 − 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.530 + 0.847i$
Analytic conductor: \(22.4601\)
Root analytic conductor: \(22.4601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 209,\ (1:\ ),\ -0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3864061533 + 0.6975939982i\)
\(L(\frac12)\) \(\approx\) \(0.3864061533 + 0.6975939982i\)
\(L(1)\) \(\approx\) \(0.5840094662 + 0.3579500421i\)
\(L(1)\) \(\approx\) \(0.5840094662 + 0.3579500421i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.0348 + 0.999i)T \)
3 \( 1 + (-0.438 + 0.898i)T \)
5 \( 1 + (-0.374 - 0.927i)T \)
7 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (-0.990 + 0.139i)T \)
17 \( 1 + (-0.615 + 0.788i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.961 + 0.275i)T \)
53 \( 1 + (0.374 - 0.927i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (0.848 - 0.529i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.374 + 0.927i)T \)
73 \( 1 + (-0.719 - 0.694i)T \)
79 \( 1 + (0.882 - 0.469i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.0348 + 0.999i)T \)
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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

−26.41282905083792656051099448780, −25.14816915963403492695457798676, −24.214769497816776086599671027917, −22.943246014248171799937632312501, −22.45631067697727028855658383639, −21.718730024500982008471715525534, −20.30243431589239076198291281219, −19.20224848724901230318737300605, −18.78304287253200516698812297583, −17.96909748074985558651098841552, −17.04905570665357049660829686683, −15.37656894277491662829790613150, −14.34196938444420556551177183386, −13.28522035376845351277150863810, −12.21941851981095205429264744971, −11.64469258231397664806496282665, −10.73995598907587597887288882904, −9.49445281782436034694993559545, −8.237029705408500301461737105180, −7.13677089326625186870499994203, −5.874267302660485455841063639770, −4.63595525322157486157667047251, −2.80608178159955492619462942653, −2.30736722620643502553013708013, −0.43376983468682440598616266164, 0.8018468728634939724824544478, 3.71670968088956349327245154119, 4.53420055107587787685461361838, 5.33810090710010933505549110127, 6.68445218261582711901537826610, 7.82018830968077674991722612420, 8.98911299162810731340282278385, 9.79317570864000930752422079490, 10.94032985908858977932332121421, 12.35331706268826136106273508982, 13.34904164779223402540368141419, 14.58509362295929818381927095895, 15.45487984323194404685467621665, 16.36921093317959113528354631649, 17.05480875705534135227114795082, 17.573830521861919799008721871547, 19.33517713236000217216944437880, 20.18089365502434427357888899766, 21.36069606421059722829682325338, 22.27233956524006146225566991998, 23.33052022699698918492378223370, 23.82703260835975824081789889952, 24.846815486799531017655950821009, 26.07345195950597438067347752013, 26.85706828749833257590595687452

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