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Properties

Label 2-133-1.1-c1-0-0
Degree 22
Conductor 133133
Sign 11
Analytic cond. 1.062011.06201
Root an. cond. 1.030531.03053
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  − 1.93·2-s + 1.25·3-s + 1.74·4-s + 1.68·5-s − 2.42·6-s − 7-s + 0.491·8-s − 1.42·9-s − 3.25·10-s + 4.93·11-s + 2.18·12-s + 1.68·13-s + 1.93·14-s + 2.10·15-s − 4.44·16-s + 5.44·17-s + 2.76·18-s + 19-s + 2.93·20-s − 1.25·21-s − 9.55·22-s + 1.81·23-s + 0.616·24-s − 2.17·25-s − 3.25·26-s − 5.55·27-s − 1.74·28-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.724·3-s + 0.872·4-s + 0.751·5-s − 0.990·6-s − 0.377·7-s + 0.173·8-s − 0.475·9-s − 1.02·10-s + 1.48·11-s + 0.632·12-s + 0.466·13-s + 0.517·14-s + 0.544·15-s − 1.11·16-s + 1.32·17-s + 0.651·18-s + 0.229·19-s + 0.656·20-s − 0.273·21-s − 2.03·22-s + 0.377·23-s + 0.125·24-s − 0.434·25-s − 0.638·26-s − 1.06·27-s − 0.329·28-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: 11
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 133, ( :1/2), 1)(2,\ 133,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.77740530910.7774053091
L(12)L(\frac12) ≈\approx 0.77740530910.7774053091
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)
bad7 1+T 1 + T
19 1−T 1 - T
good2 1+1.93T+2T2 1 + 1.93T + 2T^{2}
3 1−1.25T+3T2 1 - 1.25T + 3T^{2}
5 1−1.68T+5T2 1 - 1.68T + 5T^{2}
11 1−4.93T+11T2 1 - 4.93T + 11T^{2}
13 1−1.68T+13T2 1 - 1.68T + 13T^{2}
17 1−5.44T+17T2 1 - 5.44T + 17T^{2}
23 1−1.81T+23T2 1 - 1.81T + 23T^{2}
29 1+7.78T+29T2 1 + 7.78T + 29T^{2}
31 1+1.57T+31T2 1 + 1.57T + 31T^{2}
37 1+7.55T+37T2 1 + 7.55T + 37T^{2}
41 1+6.46T+41T2 1 + 6.46T + 41T^{2}
43 1−3.36T+43T2 1 - 3.36T + 43T^{2}
47 1+0.697T+47T2 1 + 0.697T + 47T^{2}
53 1−5.12T+53T2 1 - 5.12T + 53T^{2}
59 1+5.17T+59T2 1 + 5.17T + 59T^{2}
61 1+10.0T+61T2 1 + 10.0T + 61T^{2}
67 1+9.31T+67T2 1 + 9.31T + 67T^{2}
71 1−8.06T+71T2 1 - 8.06T + 71T^{2}
73 1−10.9T+73T2 1 - 10.9T + 73T^{2}
79 1−4.37T+79T2 1 - 4.37T + 79T^{2}
83 1−4.87T+83T2 1 - 4.87T + 83T^{2}
89 1+18.2T+89T2 1 + 18.2T + 89T^{2}
97 1+2.95T+97T2 1 + 2.95T + 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

−13.56634883368263537699149278017, −12.03823919823389430923938268075, −10.87675970720059046038391015698, −9.625349903951408654685580415956, −9.257980137925230042967412683471, −8.295451240516401268120366927673, −7.11599402563312628711147577901, −5.79213972495540220098003968585, −3.53269569005732218897252869397, −1.66395853662428755759686857237, 1.66395853662428755759686857237, 3.53269569005732218897252869397, 5.79213972495540220098003968585, 7.11599402563312628711147577901, 8.295451240516401268120366927673, 9.257980137925230042967412683471, 9.625349903951408654685580415956, 10.87675970720059046038391015698, 12.03823919823389430923938268075, 13.56634883368263537699149278017

Graph of the ZZ-function along the critical line