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Properties

Label 2-1200-1.1-c3-0-24
Degree 22
Conductor 12001200
Sign 11
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 00

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: 11
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: 00
Selberg data: (2, 1200, ( :3/2), 1)(2,\ 1200,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.3072566823.307256682
L(12)L(\frac12) ≈\approx 3.3072566823.307256682
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

−9.079087925847133918258633480504, −8.699196896808405027754966662169, −7.925664972674091914237056439637, −6.69930778026511787663650373074, −6.36647011455408362461244122054, −4.83649225388288141052162023904, −4.20660091284425316509616352167, −3.15130076087417641305182643318, −1.98046210369980688494576034578, −0.965307921502203093187681599164, 0.965307921502203093187681599164, 1.98046210369980688494576034578, 3.15130076087417641305182643318, 4.20660091284425316509616352167, 4.83649225388288141052162023904, 6.36647011455408362461244122054, 6.69930778026511787663650373074, 7.925664972674091914237056439637, 8.699196896808405027754966662169, 9.079087925847133918258633480504

Graph of the ZZ-function along the critical line