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Properties

Label 2-90e2-1.1-c1-0-33
Degree 22
Conductor 81008100
Sign 11
Analytic cond. 64.678864.6788
Root an. cond. 8.042318.04231
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  + 4·7-s − 3·11-s + 4·13-s + 5·19-s − 6·23-s + 9·29-s + 5·31-s − 2·37-s + 9·41-s + 10·43-s − 6·47-s + 9·49-s − 12·53-s − 9·59-s − 10·61-s − 2·67-s − 3·71-s + 4·73-s − 12·77-s − 4·79-s + 6·83-s + 9·89-s + 16·91-s − 2·97-s + 15·101-s + 10·103-s + 6·107-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.904·11-s + 1.10·13-s + 1.14·19-s − 1.25·23-s + 1.67·29-s + 0.898·31-s − 0.328·37-s + 1.40·41-s + 1.52·43-s − 0.875·47-s + 9/7·49-s − 1.64·53-s − 1.17·59-s − 1.28·61-s − 0.244·67-s − 0.356·71-s + 0.468·73-s − 1.36·77-s − 0.450·79-s + 0.658·83-s + 0.953·89-s + 1.67·91-s − 0.203·97-s + 1.49·101-s + 0.985·103-s + 0.580·107-s + ⋯

Functional equation

Λ(s)=(8100s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8100s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 81008100    =    22⋅34⋅522^{2} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 64.678864.6788
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8100, ( :1/2), 1)(2,\ 8100,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7367712322.736771232
L(12)L(\frac12) ≈\approx 2.7367712322.736771232
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 1−4T+pT2 1 - 4 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−9T+pT2 1 - 9 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−7.88262337731851311870417921589, −7.43465328810569102022993245115, −6.19813444562153059945956727549, −5.87078922534867975245864646985, −4.74850511007640324073209228514, −4.63150019637301685883305003179, −3.47040373748277996725817328454, −2.63796469638981412639676185280, −1.68310464171848719416573350918, −0.873605524273080761158972550445, 0.873605524273080761158972550445, 1.68310464171848719416573350918, 2.63796469638981412639676185280, 3.47040373748277996725817328454, 4.63150019637301685883305003179, 4.74850511007640324073209228514, 5.87078922534867975245864646985, 6.19813444562153059945956727549, 7.43465328810569102022993245115, 7.88262337731851311870417921589

Graph of the ZZ-function along the critical line