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Properties

Label 1-209-209.208-r0-0-0
Degree $1$
Conductor $209$
Sign $1$
Analytic cond. $0.970591$
Root an. cond. $0.970591$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 20-s + 21-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + 29-s − 30-s − 31-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 20-s + 21-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + 29-s − 30-s − 31-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.970591\)
Root analytic conductor: \(0.970591\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{209} (208, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 209,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.819007195\)
\(L(\frac12)\) \(\approx\) \(1.819007195\)
\(L(1)\) \(\approx\) \(1.582843421\)
\(L(1)\) \(\approx\) \(1.582843421\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
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

−26.52157086857984237436147729784, −25.48799545286660269029242100386, −24.77002847342185424294462072275, −23.64966656516860431172661616517, −22.86103986871479932325827835601, −22.13127744545099791727069350925, −21.42529334457695320800231971678, −20.467055439286099450472592586173, −19.15411608742692215700768213299, −18.00080019171790357813052379116, −16.95489017094725256500110424661, −16.13680605556753732982222986140, −15.31386443953591105073857867050, −13.79001820351988707356142082051, −13.120808286855691007396110208804, −12.37097673137133506530062607593, −11.040553244941788538654473940064, −10.39768940998969598821517204932, −9.10850750826991243831145318440, −7.01579200081937518330201058299, −6.32631529122611169167695912482, −5.57269004835378996106227683316, −4.40522554417283633106510990359, −3.01762053306054782097821678127, −1.50853101202667268041330150571, 1.50853101202667268041330150571, 3.01762053306054782097821678127, 4.40522554417283633106510990359, 5.57269004835378996106227683316, 6.32631529122611169167695912482, 7.01579200081937518330201058299, 9.10850750826991243831145318440, 10.39768940998969598821517204932, 11.040553244941788538654473940064, 12.37097673137133506530062607593, 13.120808286855691007396110208804, 13.79001820351988707356142082051, 15.31386443953591105073857867050, 16.13680605556753732982222986140, 16.95489017094725256500110424661, 18.00080019171790357813052379116, 19.15411608742692215700768213299, 20.467055439286099450472592586173, 21.42529334457695320800231971678, 22.13127744545099791727069350925, 22.86103986871479932325827835601, 23.64966656516860431172661616517, 24.77002847342185424294462072275, 25.48799545286660269029242100386, 26.52157086857984237436147729784

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