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Properties

Label 2-2800-1.1-c1-0-46
Degree 22
Conductor 28002800
Sign −1-1
Analytic cond. 22.358122.3581
Root an. cond. 4.728434.72843
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 − 7-s − 2·9-s + 3·11-s − 13-s − 7·17-s − 21-s + 6·23-s − 5·27-s − 5·29-s − 2·31-s + 3·33-s − 2·37-s − 39-s + 2·41-s − 4·43-s − 3·47-s + 49-s − 7·51-s − 6·53-s − 10·59-s − 8·61-s + 2·63-s + 2·67-s + 6·69-s + 8·71-s − 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s − 0.218·21-s + 1.25·23-s − 0.962·27-s − 0.928·29-s − 0.359·31-s + 0.522·33-s − 0.328·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.437·47-s + 1/7·49-s − 0.980·51-s − 0.824·53-s − 1.30·59-s − 1.02·61-s + 0.251·63-s + 0.244·67-s + 0.722·69-s + 0.949·71-s − 0.702·73-s + ⋯

Functional equation

Λ(s)=(2800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28002800    =    24⋅52⋅72^{4} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 22.358122.3581
Root analytic conductor: 4.728434.72843
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2800, ( :1/2), −1)(2,\ 2800,\ (\ :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 1
7 1+T 1 + T
good3 1−T+pT2 1 - T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+7T+pT2 1 + 7 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+8T+pT2 1 + 8 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+6T+pT2 1 + 6 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+7T+pT2 1 + 7 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

−8.610113716136751744673815020975, −7.71169390110110859831183707512, −6.84868577839913740225832418072, −6.30449045983008656645459540885, −5.29174886563521887540154085702, −4.37606541842962934803595143148, −3.48744089310847421866892408097, −2.69661210339499443876888627676, −1.68813850036432900847535927863, 0, 1.68813850036432900847535927863, 2.69661210339499443876888627676, 3.48744089310847421866892408097, 4.37606541842962934803595143148, 5.29174886563521887540154085702, 6.30449045983008656645459540885, 6.84868577839913740225832418072, 7.71169390110110859831183707512, 8.610113716136751744673815020975

Graph of the ZZ-function along the critical line