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Properties

Label 2-2268-1.1-c1-0-17
Degree 22
Conductor 22682268
Sign −1-1
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
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  − 0.239·5-s − 7-s + 5.12·11-s − 4.88·13-s − 3.70·17-s + 3.66·19-s − 7.42·23-s − 4.94·25-s − 3.46·29-s − 0.717·31-s + 0.239·35-s + 4.60·37-s − 5.60·41-s + 12.4·43-s − 4.33·47-s + 49-s + 0.942·53-s − 1.22·55-s − 7.57·59-s + 5.50·61-s + 1.16·65-s − 0.660·67-s − 13.7·71-s − 3.66·73-s − 5.12·77-s − 6.22·79-s − 9.70·83-s + ⋯
L(s)  = 1  − 0.106·5-s − 0.377·7-s + 1.54·11-s − 1.35·13-s − 0.898·17-s + 0.839·19-s − 1.54·23-s − 0.988·25-s − 0.643·29-s − 0.128·31-s + 0.0404·35-s + 0.756·37-s − 0.875·41-s + 1.90·43-s − 0.633·47-s + 0.142·49-s + 0.129·53-s − 0.165·55-s − 0.986·59-s + 0.705·61-s + 0.144·65-s − 0.0806·67-s − 1.63·71-s − 0.428·73-s − 0.584·77-s − 0.700·79-s − 1.06·83-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: −1-1
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2268, ( :1/2), −1)(2,\ 2268,\ (\ :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
7 1+T 1 + T
good5 1+0.239T+5T2 1 + 0.239T + 5T^{2}
11 1−5.12T+11T2 1 - 5.12T + 11T^{2}
13 1+4.88T+13T2 1 + 4.88T + 13T^{2}
17 1+3.70T+17T2 1 + 3.70T + 17T^{2}
19 1−3.66T+19T2 1 - 3.66T + 19T^{2}
23 1+7.42T+23T2 1 + 7.42T + 23T^{2}
29 1+3.46T+29T2 1 + 3.46T + 29T^{2}
31 1+0.717T+31T2 1 + 0.717T + 31T^{2}
37 1−4.60T+37T2 1 - 4.60T + 37T^{2}
41 1+5.60T+41T2 1 + 5.60T + 41T^{2}
43 1−12.4T+43T2 1 - 12.4T + 43T^{2}
47 1+4.33T+47T2 1 + 4.33T + 47T^{2}
53 1−0.942T+53T2 1 - 0.942T + 53T^{2}
59 1+7.57T+59T2 1 + 7.57T + 59T^{2}
61 1−5.50T+61T2 1 - 5.50T + 61T^{2}
67 1+0.660T+67T2 1 + 0.660T + 67T^{2}
71 1+13.7T+71T2 1 + 13.7T + 71T^{2}
73 1+3.66T+73T2 1 + 3.66T + 73T^{2}
79 1+6.22T+79T2 1 + 6.22T + 79T^{2}
83 1+9.70T+83T2 1 + 9.70T + 83T^{2}
89 1−7.48T+89T2 1 - 7.48T + 89T^{2}
97 1+17.1T+97T2 1 + 17.1T + 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

−8.752857055851804602153299587987, −7.73614878834370079651814229214, −7.13680425738851781114520364224, −6.30293316676175881342753178060, −5.59363285794772898477881406677, −4.39095307747203220138943377344, −3.88200947149418833363244070641, −2.67870240381194052101944133764, −1.63042204979926934887120813994, 0, 1.63042204979926934887120813994, 2.67870240381194052101944133764, 3.88200947149418833363244070641, 4.39095307747203220138943377344, 5.59363285794772898477881406677, 6.30293316676175881342753178060, 7.13680425738851781114520364224, 7.73614878834370079651814229214, 8.752857055851804602153299587987

Graph of the ZZ-function along the critical line