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Properties

Label 1-209-209.208-r0-0-0
Degree 11
Conductor 209209
Sign 11
Analytic cond. 0.9705910.970591
Root an. cond. 0.9705910.970591
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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

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

Invariants

Degree: 11
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: 11
Analytic conductor: 0.9705910.970591
Root analytic conductor: 0.9705910.970591
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ209(208,⋅)\chi_{209} (208, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 209, (0: ), 1)(1,\ 209,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8190071951.819007195
L(12)L(\frac12) ≈\approx 1.8190071951.819007195
L(1)L(1) ≈\approx 1.5828434211.582843421
L(1)L(1) ≈\approx 1.5828434211.582843421

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad11 1 1
19 1 1
good2 1+T 1 + T
3 1−T 1 - T
5 1+T 1 + T
7 1−T 1 - T
13 1+T 1 + T
17 1−T 1 - T
23 1+T 1 + T
29 1+T 1 + T
31 1−T 1 - T
37 1−T 1 - T
41 1+T 1 + T
43 1−T 1 - T
47 1+T 1 + T
53 1−T 1 - T
59 1−T 1 - T
61 1−T 1 - T
67 1−T 1 - T
71 1−T 1 - T
73 1−T 1 - T
79 1+T 1 + T
83 1−T 1 - T
89 1−T 1 - T
97 1−T 1 - T
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   L(s)=∏p (1−αp p−s)−1L(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 ZZ-function along the critical line