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Properties

Label 2-90e2-1.1-c1-0-69
Degree 22
Conductor 81008100
Sign −1-1
Analytic cond. 64.678864.6788
Root an. cond. 8.042318.04231
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.73·7-s + 1.73·11-s − 5.46·13-s + 4.73·17-s − 4.46·19-s − 3.46·23-s − 7.73·29-s + 5.92·31-s + 6.19·37-s − 11.1·41-s − 3.26·43-s + 1.26·47-s + 0.464·49-s + 7.26·53-s − 7.73·59-s − 4·61-s − 6.39·67-s − 11.1·71-s + 0.196·73-s + 4.73·77-s − 14.3·79-s + 15.1·83-s + 5.19·89-s − 14.9·91-s − 0.732·97-s + 6.12·101-s − 18.3·103-s + ⋯
L(s)  = 1  + 1.03·7-s + 0.522·11-s − 1.51·13-s + 1.14·17-s − 1.02·19-s − 0.722·23-s − 1.43·29-s + 1.06·31-s + 1.01·37-s − 1.74·41-s − 0.498·43-s + 0.184·47-s + 0.0663·49-s + 0.998·53-s − 1.00·59-s − 0.512·61-s − 0.780·67-s − 1.32·71-s + 0.0229·73-s + 0.539·77-s − 1.61·79-s + 1.66·83-s + 0.550·89-s − 1.56·91-s − 0.0743·97-s + 0.609·101-s − 1.81·103-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: −1-1
Analytic conductor: 64.678864.6788
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8100, ( :1/2), −1)(2,\ 8100,\ (\ :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 1−2.73T+7T2 1 - 2.73T + 7T^{2}
11 1−1.73T+11T2 1 - 1.73T + 11T^{2}
13 1+5.46T+13T2 1 + 5.46T + 13T^{2}
17 1−4.73T+17T2 1 - 4.73T + 17T^{2}
19 1+4.46T+19T2 1 + 4.46T + 19T^{2}
23 1+3.46T+23T2 1 + 3.46T + 23T^{2}
29 1+7.73T+29T2 1 + 7.73T + 29T^{2}
31 1−5.92T+31T2 1 - 5.92T + 31T^{2}
37 1−6.19T+37T2 1 - 6.19T + 37T^{2}
41 1+11.1T+41T2 1 + 11.1T + 41T^{2}
43 1+3.26T+43T2 1 + 3.26T + 43T^{2}
47 1−1.26T+47T2 1 - 1.26T + 47T^{2}
53 1−7.26T+53T2 1 - 7.26T + 53T^{2}
59 1+7.73T+59T2 1 + 7.73T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1+6.39T+67T2 1 + 6.39T + 67T^{2}
71 1+11.1T+71T2 1 + 11.1T + 71T^{2}
73 1−0.196T+73T2 1 - 0.196T + 73T^{2}
79 1+14.3T+79T2 1 + 14.3T + 79T^{2}
83 1−15.1T+83T2 1 - 15.1T + 83T^{2}
89 1−5.19T+89T2 1 - 5.19T + 89T^{2}
97 1+0.732T+97T2 1 + 0.732T + 97T^{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

−7.64492934030562224390294336017, −6.86860471143552405135412391966, −6.05178237762916028303743155503, −5.32265686337292140703638484320, −4.65262536782745171118848489561, −4.06047142137456102857040186837, −3.04342752116821522881340923307, −2.11734918491584568021209460398, −1.40634830012377957720142661344, 0, 1.40634830012377957720142661344, 2.11734918491584568021209460398, 3.04342752116821522881340923307, 4.06047142137456102857040186837, 4.65262536782745171118848489561, 5.32265686337292140703638484320, 6.05178237762916028303743155503, 6.86860471143552405135412391966, 7.64492934030562224390294336017

Graph of the ZZ-function along the critical line