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Properties

Label 2-1800-1.1-c1-0-18
Degree 22
Conductor 18001800
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 4·11-s − 4·13-s − 4·19-s − 2·23-s − 2·29-s − 4·37-s − 2·41-s + 6·43-s − 6·47-s − 3·49-s − 4·53-s + 12·59-s − 10·61-s − 14·67-s − 8·71-s − 8·73-s − 8·77-s + 16·79-s + 2·83-s − 6·89-s + 8·91-s − 16·97-s − 6·101-s − 14·103-s − 10·107-s − 6·109-s + ⋯
L(s)  = 1  − 0.755·7-s + 1.20·11-s − 1.10·13-s − 0.917·19-s − 0.417·23-s − 0.371·29-s − 0.657·37-s − 0.312·41-s + 0.914·43-s − 0.875·47-s − 3/7·49-s − 0.549·53-s + 1.56·59-s − 1.28·61-s − 1.71·67-s − 0.949·71-s − 0.936·73-s − 0.911·77-s + 1.80·79-s + 0.219·83-s − 0.635·89-s + 0.838·91-s − 1.62·97-s − 0.597·101-s − 1.37·103-s − 0.966·107-s − 0.574·109-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: −1-1
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: 11
Selberg data: (2, 1800, ( :1/2), −1)(2,\ 1800,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−4T+pT2 1 - 4 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+14T+pT2 1 + 14 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+16T+pT2 1 + 16 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

−9.046460112719450832914573677804, −8.125314960122643492265723439737, −7.14831674158858802892659992469, −6.55536659882703175830371730918, −5.77630218563910465834928061038, −4.65101662855925806243412279834, −3.86307675984348758158108701206, −2.85591890992545130759706287184, −1.69174310740343087998151716036, 0, 1.69174310740343087998151716036, 2.85591890992545130759706287184, 3.86307675984348758158108701206, 4.65101662855925806243412279834, 5.77630218563910465834928061038, 6.55536659882703175830371730918, 7.14831674158858802892659992469, 8.125314960122643492265723439737, 9.046460112719450832914573677804

Graph of the ZZ-function along the critical line