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Properties

Label 2-3822-1.1-c1-0-71
Degree 22
Conductor 38223822
Sign −1-1
Analytic cond. 30.518830.5188
Root an. cond. 5.524385.52438
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 15-s + 16-s − 5·17-s − 18-s + 19-s + 20-s − 22-s − 5·23-s − 24-s − 4·25-s + 26-s + 27-s − 3·29-s − 30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.213·22-s − 1.04·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.557·29-s − 0.182·30-s + 0.359·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(3822s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3822s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38223822    =    2⋅3⋅72⋅132 \cdot 3 \cdot 7^{2} \cdot 13
Sign: −1-1
Analytic conductor: 30.518830.5188
Root analytic conductor: 5.524385.52438
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3822, ( :1/2), −1)(2,\ 3822,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1+T 1 + T
3 1−T 1 - T
7 1 1
13 1+T 1 + T
good5 1−T+pT2 1 - T + p T^{2}
11 1−T+pT2 1 - T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+5T+pT2 1 + 5 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+11T+pT2 1 + 11 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1−T+pT2 1 - T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{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

−8.229624320832690142984378383684, −7.52816032351613775475960337785, −6.75482774895766807647838084195, −6.13224739380455376431007248026, −5.15918727573138497040371551761, −4.16478786226631774399464294051, −3.28562902398430010868422011747, −2.21183611182006506064580779862, −1.64319459172741036695391272168, 0, 1.64319459172741036695391272168, 2.21183611182006506064580779862, 3.28562902398430010868422011747, 4.16478786226631774399464294051, 5.15918727573138497040371551761, 6.13224739380455376431007248026, 6.75482774895766807647838084195, 7.52816032351613775475960337785, 8.229624320832690142984378383684

Graph of the ZZ-function along the critical line