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Properties

Label 2-1200-1.1-c3-0-37
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 − 10·7-s + 9·9-s + 46·11-s − 34·13-s + 66·17-s − 104·19-s + 30·21-s − 164·23-s − 27·27-s + 224·29-s + 72·31-s − 138·33-s − 22·37-s + 102·39-s + 194·41-s − 108·43-s + 480·47-s − 243·49-s − 198·51-s + 286·53-s + 312·57-s − 426·59-s + 698·61-s − 90·63-s − 328·67-s + 492·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.539·7-s + 1/3·9-s + 1.26·11-s − 0.725·13-s + 0.941·17-s − 1.25·19-s + 0.311·21-s − 1.48·23-s − 0.192·27-s + 1.43·29-s + 0.417·31-s − 0.727·33-s − 0.0977·37-s + 0.418·39-s + 0.738·41-s − 0.383·43-s + 1.48·47-s − 0.708·49-s − 0.543·51-s + 0.741·53-s + 0.725·57-s − 0.940·59-s + 1.46·61-s − 0.179·63-s − 0.598·67-s + 0.858·69-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+10T+p3T2 1 + 10 T + p^{3} T^{2}
11 1−46T+p3T2 1 - 46 T + p^{3} T^{2}
13 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
17 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
19 1+104T+p3T2 1 + 104 T + p^{3} T^{2}
23 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
29 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
31 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
37 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
41 1−194T+p3T2 1 - 194 T + p^{3} T^{2}
43 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
47 1−480T+p3T2 1 - 480 T + p^{3} T^{2}
53 1−286T+p3T2 1 - 286 T + p^{3} T^{2}
59 1+426T+p3T2 1 + 426 T + p^{3} T^{2}
61 1−698T+p3T2 1 - 698 T + p^{3} T^{2}
67 1+328T+p3T2 1 + 328 T + p^{3} T^{2}
71 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
73 1+740T+p3T2 1 + 740 T + p^{3} T^{2}
79 1+1168T+p3T2 1 + 1168 T + p^{3} T^{2}
83 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
89 1−1206T+p3T2 1 - 1206 T + p^{3} T^{2}
97 1+1384T+p3T2 1 + 1384 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

−9.041056047962812988723373276776, −8.158742672835397609184854016065, −7.13797839556417246695348517062, −6.37762354123565371483459210891, −5.78360378119950164451846780587, −4.54211796593548807409055654972, −3.85791401515934591379026062499, −2.55904891088663822032084220208, −1.25302284592991174915457073030, 0, 1.25302284592991174915457073030, 2.55904891088663822032084220208, 3.85791401515934591379026062499, 4.54211796593548807409055654972, 5.78360378119950164451846780587, 6.37762354123565371483459210891, 7.13797839556417246695348517062, 8.158742672835397609184854016065, 9.041056047962812988723373276776

Graph of the ZZ-function along the critical line