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Properties

Label 2-1520-1.1-c1-0-22
Degree 22
Conductor 15201520
Sign −1-1
Analytic cond. 12.137212.1372
Root an. cond. 3.483853.48385
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  − 2·3-s + 5-s − 4·7-s + 9-s + 4·11-s − 2·15-s + 6·17-s + 19-s + 8·21-s − 8·23-s + 25-s + 4·27-s − 6·29-s + 8·31-s − 8·33-s − 4·35-s − 8·37-s − 2·41-s + 45-s − 12·47-s + 9·49-s − 12·51-s + 4·53-s + 4·55-s − 2·57-s − 8·59-s − 14·61-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 1.45·17-s + 0.229·19-s + 1.74·21-s − 1.66·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 1.43·31-s − 1.39·33-s − 0.676·35-s − 1.31·37-s − 0.312·41-s + 0.149·45-s − 1.75·47-s + 9/7·49-s − 1.68·51-s + 0.549·53-s + 0.539·55-s − 0.264·57-s − 1.04·59-s − 1.79·61-s + ⋯

Functional equation

Λ(s)=(1520s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1520s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15201520    =    24⋅5⋅192^{4} \cdot 5 \cdot 19
Sign: −1-1
Analytic conductor: 12.137212.1372
Root analytic conductor: 3.483853.48385
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1520, ( :1/2), −1)(2,\ 1520,\ (\ :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
19 1−T 1 - T
good3 1+2T+pT2 1 + 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+pT2 1 + 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

−9.414389171153941052152571176806, −8.306558588442832701284063292788, −7.15092239567899718438965748252, −6.26604533373139617436495529729, −6.07553649243771927211326383485, −5.14236160489505260986626044288, −3.89039366794407092271004025254, −3.07425032684048770058081362284, −1.43164889163469816022501322949, 0, 1.43164889163469816022501322949, 3.07425032684048770058081362284, 3.89039366794407092271004025254, 5.14236160489505260986626044288, 6.07553649243771927211326383485, 6.26604533373139617436495529729, 7.15092239567899718438965748252, 8.306558588442832701284063292788, 9.414389171153941052152571176806

Graph of the ZZ-function along the critical line