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Properties

Label 2-1200-1.1-c3-0-54
Degree 22
Conductor 12001200
Sign −1-1
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
Motivic weight 33
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 + 13·7-s + 9·9-s − 6·11-s + 5·13-s − 78·17-s − 65·19-s + 39·21-s − 138·23-s + 27·27-s + 66·29-s − 299·31-s − 18·33-s − 214·37-s + 15·39-s + 360·41-s − 203·43-s − 78·47-s − 174·49-s − 234·51-s + 636·53-s − 195·57-s − 786·59-s + 467·61-s + 117·63-s + 217·67-s − 414·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.701·7-s + 1/3·9-s − 0.164·11-s + 0.106·13-s − 1.11·17-s − 0.784·19-s + 0.405·21-s − 1.25·23-s + 0.192·27-s + 0.422·29-s − 1.73·31-s − 0.0949·33-s − 0.950·37-s + 0.0615·39-s + 1.37·41-s − 0.719·43-s − 0.242·47-s − 0.507·49-s − 0.642·51-s + 1.64·53-s − 0.453·57-s − 1.73·59-s + 0.980·61-s + 0.233·63-s + 0.395·67-s − 0.722·69-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 70.802270.8022
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1200, ( :3/2), −1)(2,\ 1200,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
3 1−pT 1 - p T
5 1 1
good7 1−13T+p3T2 1 - 13 T + p^{3} T^{2}
11 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
13 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
17 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
19 1+65T+p3T2 1 + 65 T + p^{3} T^{2}
23 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
29 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
31 1+299T+p3T2 1 + 299 T + p^{3} T^{2}
37 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
41 1−360T+p3T2 1 - 360 T + p^{3} T^{2}
43 1+203T+p3T2 1 + 203 T + p^{3} T^{2}
47 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
53 1−12pT+p3T2 1 - 12 p T + p^{3} T^{2}
59 1+786T+p3T2 1 + 786 T + p^{3} T^{2}
61 1−467T+p3T2 1 - 467 T + p^{3} T^{2}
67 1−217T+p3T2 1 - 217 T + p^{3} T^{2}
71 1−360T+p3T2 1 - 360 T + p^{3} T^{2}
73 1+286T+p3T2 1 + 286 T + p^{3} T^{2}
79 1+272T+p3T2 1 + 272 T + p^{3} T^{2}
83 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
89 1+p3T2 1 + p^{3} T^{2}
97 1+511T+p3T2 1 + 511 T + p^{3} 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

−8.803500134319591274932351104780, −8.278728828930639424910110952894, −7.42691537302626434831512982401, −6.56105726080252807587063328287, −5.52739754439847007169892125460, −4.49560752754506680873337117645, −3.75979948387020228305105931149, −2.42930041771837526207141840307, −1.66105556554310891302434905547, 0, 1.66105556554310891302434905547, 2.42930041771837526207141840307, 3.75979948387020228305105931149, 4.49560752754506680873337117645, 5.52739754439847007169892125460, 6.56105726080252807587063328287, 7.42691537302626434831512982401, 8.278728828930639424910110952894, 8.803500134319591274932351104780

Graph of the ZZ-function along the critical line