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Properties

Label 2-1800-1.1-c1-0-1
Degree 22
Conductor 18001800
Sign 11
Analytic cond. 14.373014.3730
Root an. cond. 3.791183.79118
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 2·11-s − 4·13-s − 2·17-s + 4·19-s + 8·23-s + 10·29-s + 4·31-s + 8·43-s + 8·47-s − 3·49-s + 6·53-s + 14·59-s − 14·61-s + 4·67-s − 12·71-s − 6·73-s + 4·77-s − 12·79-s + 4·83-s + 12·89-s + 8·91-s + 14·97-s − 6·101-s + 14·103-s − 12·107-s + 2·109-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.603·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1.85·29-s + 0.718·31-s + 1.21·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 1.82·59-s − 1.79·61-s + 0.488·67-s − 1.42·71-s − 0.702·73-s + 0.455·77-s − 1.35·79-s + 0.439·83-s + 1.27·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s + 1.37·103-s − 1.16·107-s + 0.191·109-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 14.373014.3730
Root analytic conductor: 3.791183.79118
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1800, ( :1/2), 1)(2,\ 1800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4033381591.403338159
L(12)L(\frac12) ≈\approx 1.4033381591.403338159
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
3 1 1
5 1 1
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−9.270407886951524481778316794857, −8.615181679584005774951790009650, −7.51380370959353581607795133194, −7.02360313048233502050208617514, −6.10428552516702050498345256322, −5.13249432243264413574946933168, −4.45417944745640591028127050063, −3.08658133002906810213405205063, −2.55273956376703116430496016927, −0.797276142181345148530396500843, 0.797276142181345148530396500843, 2.55273956376703116430496016927, 3.08658133002906810213405205063, 4.45417944745640591028127050063, 5.13249432243264413574946933168, 6.10428552516702050498345256322, 7.02360313048233502050208617514, 7.51380370959353581607795133194, 8.615181679584005774951790009650, 9.270407886951524481778316794857

Graph of the ZZ-function along the critical line