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Properties

Label 2-1200-1.1-c3-0-51
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 + 2·7-s + 9·9-s − 70·11-s + 54·13-s − 22·17-s − 24·19-s + 6·21-s + 100·23-s + 27·27-s + 216·29-s − 208·31-s − 210·33-s − 254·37-s + 162·39-s − 206·41-s − 292·43-s + 320·47-s − 339·49-s − 66·51-s − 402·53-s − 72·57-s + 370·59-s − 550·61-s + 18·63-s − 728·67-s + 300·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.107·7-s + 1/3·9-s − 1.91·11-s + 1.15·13-s − 0.313·17-s − 0.289·19-s + 0.0623·21-s + 0.906·23-s + 0.192·27-s + 1.38·29-s − 1.20·31-s − 1.10·33-s − 1.12·37-s + 0.665·39-s − 0.784·41-s − 1.03·43-s + 0.993·47-s − 0.988·49-s − 0.181·51-s − 1.04·53-s − 0.167·57-s + 0.816·59-s − 1.15·61-s + 0.0359·63-s − 1.32·67-s + 0.523·69-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 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
11 1+70T+p3T2 1 + 70 T + p^{3} T^{2}
13 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
17 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
19 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
23 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
29 1−216T+p3T2 1 - 216 T + p^{3} T^{2}
31 1+208T+p3T2 1 + 208 T + p^{3} T^{2}
37 1+254T+p3T2 1 + 254 T + p^{3} T^{2}
41 1+206T+p3T2 1 + 206 T + p^{3} T^{2}
43 1+292T+p3T2 1 + 292 T + p^{3} T^{2}
47 1−320T+p3T2 1 - 320 T + p^{3} T^{2}
53 1+402T+p3T2 1 + 402 T + p^{3} T^{2}
59 1−370T+p3T2 1 - 370 T + p^{3} T^{2}
61 1+550T+p3T2 1 + 550 T + p^{3} T^{2}
67 1+728T+p3T2 1 + 728 T + p^{3} T^{2}
71 1−540T+p3T2 1 - 540 T + p^{3} T^{2}
73 1−604T+p3T2 1 - 604 T + p^{3} T^{2}
79 1+792T+p3T2 1 + 792 T + p^{3} T^{2}
83 1+404T+p3T2 1 + 404 T + p^{3} T^{2}
89 1+938T+p3T2 1 + 938 T + p^{3} T^{2}
97 1−56T+p3T2 1 - 56 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

−8.694070151637486189019718805360, −8.332764124657891939209909511297, −7.43238125108599296220671629479, −6.56221507968919661873361177847, −5.44763920086435595426136820589, −4.70669791811104319580596608049, −3.47101906263729083197546502151, −2.68936112318262079800782738959, −1.53262138606236513808992856139, 0, 1.53262138606236513808992856139, 2.68936112318262079800782738959, 3.47101906263729083197546502151, 4.70669791811104319580596608049, 5.44763920086435595426136820589, 6.56221507968919661873361177847, 7.43238125108599296220671629479, 8.332764124657891939209909511297, 8.694070151637486189019718805360

Graph of the ZZ-function along the critical line