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Properties

Label 1-209-209.73-r1-0-0
Degree $1$
Conductor $209$
Sign $-0.992 - 0.119i$
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  + (0.882 − 0.469i)2-s + (−0.997 + 0.0697i)3-s + (0.559 − 0.829i)4-s + (−0.615 + 0.788i)5-s + (−0.848 + 0.529i)6-s + (0.104 + 0.994i)7-s + (0.104 − 0.994i)8-s + (0.990 − 0.139i)9-s + (−0.173 + 0.984i)10-s + (−0.5 + 0.866i)12-s + (0.374 − 0.927i)13-s + (0.559 + 0.829i)14-s + (0.559 − 0.829i)15-s + (−0.374 − 0.927i)16-s + (−0.990 − 0.139i)17-s + (0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)2-s + (−0.997 + 0.0697i)3-s + (0.559 − 0.829i)4-s + (−0.615 + 0.788i)5-s + (−0.848 + 0.529i)6-s + (0.104 + 0.994i)7-s + (0.104 − 0.994i)8-s + (0.990 − 0.139i)9-s + (−0.173 + 0.984i)10-s + (−0.5 + 0.866i)12-s + (0.374 − 0.927i)13-s + (0.559 + 0.829i)14-s + (0.559 − 0.829i)15-s + (−0.374 − 0.927i)16-s + (−0.990 − 0.139i)17-s + (0.809 − 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.992 - 0.119i$
Analytic conductor: \(22.4601\)
Root analytic conductor: \(22.4601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 209,\ (1:\ ),\ -0.992 - 0.119i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02672731429 - 0.4443552321i\)
\(L(\frac12)\) \(\approx\) \(0.02672731429 - 0.4443552321i\)
\(L(1)\) \(\approx\) \(0.9178281649 - 0.2091179977i\)
\(L(1)\) \(\approx\) \(0.9178281649 - 0.2091179977i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.882 - 0.469i)T \)
3 \( 1 + (-0.997 + 0.0697i)T \)
5 \( 1 + (-0.615 + 0.788i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.374 - 0.927i)T \)
17 \( 1 + (-0.990 - 0.139i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.438 - 0.898i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.997 - 0.0697i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.719 + 0.694i)T \)
53 \( 1 + (-0.615 - 0.788i)T \)
59 \( 1 + (-0.719 - 0.694i)T \)
61 \( 1 + (-0.0348 - 0.999i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.615 + 0.788i)T \)
73 \( 1 + (0.241 - 0.970i)T \)
79 \( 1 + (-0.848 - 0.529i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.882 + 0.469i)T \)
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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

−26.8701529357573424831848790422, −26.01307140819782962711440135290, −24.43160914891580341060270383913, −24.03261794016562947865069436510, −23.34309485457739338757445413960, −22.504849984562353601921488068698, −21.48540424557955785934153759276, −20.53705866557127423357225222132, −19.623807828424670233579901083976, −18.01959340787283889955998354069, −17.01701856127888299634945586935, −16.33560097836980150138226142273, −15.73486832971985278278994346949, −14.29307657567466793429799520164, −13.22844379808036793588332007796, −12.45720774626457486909420848263, −11.461440663973590607414315226773, −10.728417698661676204503820005831, −8.90662501630956153351688052086, −7.55003492209996098933690001010, −6.7908770467831463248209423380, −5.58929692181022948231718450138, −4.40978345821659264453468381702, −3.9557169435417077439981231816, −1.60788433164578216603746888352, 0.12198113238438133508437423169, 2.020938279225062687197887251441, 3.36707491646938077196414437741, 4.54190113014829086482869692269, 5.73253927789901494310712793387, 6.426985726903979753345186235076, 7.74193658486161387049472937288, 9.596594891312039989787345328968, 10.80125136354735211746730282505, 11.35916070600102434742685105488, 12.235367280081726441236020669, 13.123608430200665539413900527422, 14.53502229664230638880964808, 15.59747472000214883417641379483, 15.84321644437322457061254031940, 17.70337084041329232310202524390, 18.480636746020974309674326527179, 19.37208791366555920304754362400, 20.57617520191343220603416195111, 21.717746064171809877795989913, 22.38170673105661967860109696838, 22.87903391769799342236450492303, 23.96997184722631054350591328339, 24.66943543320751544091082734119, 26.00423714801372512458180012893

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