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Properties

Label 2-920-1.1-c1-0-13
Degree 22
Conductor 920920
Sign −1-1
Analytic cond. 7.346237.34623
Root an. cond. 2.710392.71039
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  − 3·3-s + 5-s − 2·7-s + 6·9-s + 13-s − 3·15-s + 6·21-s + 23-s + 25-s − 9·27-s − 3·29-s + 3·31-s − 2·35-s − 8·37-s − 3·39-s + 3·41-s − 2·43-s + 6·45-s − 11·47-s − 3·49-s − 14·53-s − 8·59-s − 4·61-s − 12·63-s + 65-s − 4·67-s − 3·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s − 0.755·7-s + 2·9-s + 0.277·13-s − 0.774·15-s + 1.30·21-s + 0.208·23-s + 1/5·25-s − 1.73·27-s − 0.557·29-s + 0.538·31-s − 0.338·35-s − 1.31·37-s − 0.480·39-s + 0.468·41-s − 0.304·43-s + 0.894·45-s − 1.60·47-s − 3/7·49-s − 1.92·53-s − 1.04·59-s − 0.512·61-s − 1.51·63-s + 0.124·65-s − 0.488·67-s − 0.361·69-s + ⋯

Functional equation

Λ(s)=(920s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(920s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 920920    =    23⋅5⋅232^{3} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 7.346237.34623
Root analytic conductor: 2.710392.71039
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 920, ( :1/2), −1)(2,\ 920,\ (\ :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 1
5 1−T 1 - T
23 1−T 1 - T
good3 1+pT+pT2 1 + p T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+11T+pT2 1 + 11 T + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1+9T+pT2 1 + 9 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−18T+pT2 1 - 18 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

−9.890882640173503188502898887273, −9.111657690465418843613202394863, −7.76060357445880650356822288152, −6.63403416528712923600421000252, −6.29492395479034476582513996003, −5.37961484897365172295492227072, −4.60783226407171159669213174854, −3.29362648392513345292182959612, −1.52953635996900865734002074563, 0, 1.52953635996900865734002074563, 3.29362648392513345292182959612, 4.60783226407171159669213174854, 5.37961484897365172295492227072, 6.29492395479034476582513996003, 6.63403416528712923600421000252, 7.76060357445880650356822288152, 9.111657690465418843613202394863, 9.890882640173503188502898887273

Graph of the ZZ-function along the critical line