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Properties

Label 2-209-1.1-c1-0-2
Degree 22
Conductor 209209
Sign 11
Analytic cond. 1.668871.66887
Root an. cond. 1.291841.29184
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.456·2-s − 0.835·3-s − 1.79·4-s + 0.221·5-s + 0.381·6-s + 4.69·7-s + 1.73·8-s − 2.30·9-s − 0.101·10-s − 11-s + 1.49·12-s + 5.89·13-s − 2.14·14-s − 0.185·15-s + 2.79·16-s + 7.06·17-s + 1.05·18-s + 19-s − 0.397·20-s − 3.92·21-s + 0.456·22-s + 1.06·23-s − 1.44·24-s − 4.95·25-s − 2.68·26-s + 4.42·27-s − 8.41·28-s + ⋯
L(s)  = 1  − 0.322·2-s − 0.482·3-s − 0.895·4-s + 0.0992·5-s + 0.155·6-s + 1.77·7-s + 0.612·8-s − 0.767·9-s − 0.0320·10-s − 0.301·11-s + 0.431·12-s + 1.63·13-s − 0.573·14-s − 0.0478·15-s + 0.698·16-s + 1.71·17-s + 0.247·18-s + 0.229·19-s − 0.0889·20-s − 0.856·21-s + 0.0973·22-s + 0.221·23-s − 0.295·24-s − 0.990·25-s − 0.527·26-s + 0.852·27-s − 1.59·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: 11
Analytic conductor: 1.668871.66887
Root analytic conductor: 1.291841.29184
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 209, ( :1/2), 1)(2,\ 209,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88451422380.8845142238
L(12)L(\frac12) ≈\approx 0.88451422380.8845142238
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad11 1+T 1 + T
19 1−T 1 - T
good2 1+0.456T+2T2 1 + 0.456T + 2T^{2}
3 1+0.835T+3T2 1 + 0.835T + 3T^{2}
5 1−0.221T+5T2 1 - 0.221T + 5T^{2}
7 1−4.69T+7T2 1 - 4.69T + 7T^{2}
13 1−5.89T+13T2 1 - 5.89T + 13T^{2}
17 1−7.06T+17T2 1 - 7.06T + 17T^{2}
23 1−1.06T+23T2 1 - 1.06T + 23T^{2}
29 1+7.62T+29T2 1 + 7.62T + 29T^{2}
31 1−0.901T+31T2 1 - 0.901T + 31T^{2}
37 1+2.71T+37T2 1 + 2.71T + 37T^{2}
41 1−0.788T+41T2 1 - 0.788T + 41T^{2}
43 1−0.714T+43T2 1 - 0.714T + 43T^{2}
47 1−3.96T+47T2 1 - 3.96T + 47T^{2}
53 1+9.69T+53T2 1 + 9.69T + 53T^{2}
59 1+7.33T+59T2 1 + 7.33T + 59T^{2}
61 1−8.15T+61T2 1 - 8.15T + 61T^{2}
67 1−7.86T+67T2 1 - 7.86T + 67T^{2}
71 1+3.13T+71T2 1 + 3.13T + 71T^{2}
73 1+6.49T+73T2 1 + 6.49T + 73T^{2}
79 1−12.0T+79T2 1 - 12.0T + 79T^{2}
83 1−16.3T+83T2 1 - 16.3T + 83T^{2}
89 1+12.9T+89T2 1 + 12.9T + 89T^{2}
97 1−8.14T+97T2 1 - 8.14T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.19523134593506336933721136825, −11.24315590331099936105207434402, −10.63960423743970911078648786137, −9.319308212683367048424991081905, −8.260618365977527678505227769444, −7.80739518562844672234277678145, −5.74970565770299766656927693801, −5.17842296779522439176261900762, −3.76566986960473015694358687349, −1.32305777517800720586171586944, 1.32305777517800720586171586944, 3.76566986960473015694358687349, 5.17842296779522439176261900762, 5.74970565770299766656927693801, 7.80739518562844672234277678145, 8.260618365977527678505227769444, 9.319308212683367048424991081905, 10.63960423743970911078648786137, 11.24315590331099936105207434402, 12.19523134593506336933721136825

Graph of the ZZ-function along the critical line