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Properties

Label 2-920-1.1-c1-0-19
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-s − 5-s − 2·7-s − 2·9-s + 13-s − 15-s − 4·17-s − 4·19-s − 2·21-s + 23-s + 25-s − 5·27-s − 3·29-s − 31-s + 2·35-s − 8·37-s + 39-s − 5·41-s − 6·43-s + 2·45-s + 9·47-s − 3·49-s − 4·51-s + 2·53-s − 4·57-s + 4·63-s − 65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.277·13-s − 0.258·15-s − 0.970·17-s − 0.917·19-s − 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 0.179·31-s + 0.338·35-s − 1.31·37-s + 0.160·39-s − 0.780·41-s − 0.914·43-s + 0.298·45-s + 1.31·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s − 0.529·57-s + 0.503·63-s − 0.124·65-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−T+pT2 1 - 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+4T+pT2 1 + 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+5T+pT2 1 + 5 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−9.478758761073904215941317036848, −8.755728696260485632940986674078, −8.211346506282102834214316168774, −7.07026939594301482653796233966, −6.35976630621235922374887406783, −5.26343851670073918515221325937, −4.01358333286914915727130091103, −3.23338502943497628525350311323, −2.12946958546745955907702459510, 0, 2.12946958546745955907702459510, 3.23338502943497628525350311323, 4.01358333286914915727130091103, 5.26343851670073918515221325937, 6.35976630621235922374887406783, 7.07026939594301482653796233966, 8.211346506282102834214316168774, 8.755728696260485632940986674078, 9.478758761073904215941317036848

Graph of the ZZ-function along the critical line