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Properties

Label 2-83-1.1-c1-0-6
Degree 22
Conductor 8383
Sign −1-1
Analytic cond. 0.6627580.662758
Root an. cond. 0.8140990.814099
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 − 2·5-s + 6-s − 3·7-s + 3·8-s − 2·9-s + 2·10-s + 3·11-s + 12-s − 6·13-s + 3·14-s + 2·15-s − 16-s + 5·17-s + 2·18-s + 2·19-s + 2·20-s + 3·21-s − 3·22-s − 4·23-s − 3·24-s − 25-s + 6·26-s + 5·27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.66·13-s + 0.801·14-s + 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.654·21-s − 0.639·22-s − 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.566·28-s + ⋯

Functional equation

Λ(s)=(83s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(83s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8383
Sign: −1-1
Analytic conductor: 0.6627580.662758
Root analytic conductor: 0.8140990.814099
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 83, ( :1/2), −1)(2,\ 83,\ (\ :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)
bad83 1+T 1 + T
good2 1+T+pT2 1 + T + p T^{2}
3 1+T+pT2 1 + T + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−5T+pT2 1 - 5 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+7T+pT2 1 + 7 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+11T+pT2 1 + 11 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−5T+pT2 1 - 5 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+8T+pT2 1 + 8 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

−13.82349410001976328687202362359, −12.30534590659785910801180963406, −11.77890434965261413314394451050, −10.17676635448238491848838037425, −9.457827669916618767543831706200, −8.118927993931557797577222055683, −6.97450299304430271706246705192, −5.31327024180315445602831381888, −3.65475901244453246589839654026, 0, 3.65475901244453246589839654026, 5.31327024180315445602831381888, 6.97450299304430271706246705192, 8.118927993931557797577222055683, 9.457827669916618767543831706200, 10.17676635448238491848838037425, 11.77890434965261413314394451050, 12.30534590659785910801180963406, 13.82349410001976328687202362359

Graph of the ZZ-function along the critical line