↑

Properties

Label 2-14896-1.1-c1-0-9
Degree 22
Conductor 1489614896
Sign 11
Analytic cond. 118.945118.945
Root an. cond. 10.906110.9061
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s + 6·11-s − 5·13-s − 3·17-s + 19-s − 3·23-s − 5·25-s − 5·27-s + 9·29-s − 4·31-s + 6·33-s + 2·37-s − 5·39-s − 8·43-s − 3·51-s − 3·53-s + 57-s + 9·59-s + 10·61-s − 5·67-s − 3·69-s + 6·71-s + 7·73-s − 5·75-s + 10·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 1.80·11-s − 1.38·13-s − 0.727·17-s + 0.229·19-s − 0.625·23-s − 25-s − 0.962·27-s + 1.67·29-s − 0.718·31-s + 1.04·33-s + 0.328·37-s − 0.800·39-s − 1.21·43-s − 0.420·51-s − 0.412·53-s + 0.132·57-s + 1.17·59-s + 1.28·61-s − 0.610·67-s − 0.361·69-s + 0.712·71-s + 0.819·73-s − 0.577·75-s + 1.12·79-s + 1/9·81-s + ⋯

Functional equation

Λ(s)=(14896s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 14896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(14896s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1489614896    =    24⋅72⋅192^{4} \cdot 7^{2} \cdot 19
Sign: 11
Analytic conductor: 118.945118.945
Root analytic conductor: 10.906110.9061
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 14896, ( :1/2), 1)(2,\ 14896,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1437754432.143775443
L(12)L(\frac12) ≈\approx 2.1437754432.143775443
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
7 1 1
19 1−T 1 - T
good3 1−T+pT2 1 - T + p T^{2}
5 1+pT2 1 + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+3T+pT2 1 + 3 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+5T+pT2 1 + 5 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{2}
show more
show less
   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

−16.05710859748804, −15.37174194663751, −14.79935216780154, −14.37789233563134, −14.01621114862507, −13.46000282716391, −12.65482892440077, −11.97007488543712, −11.70331043719405, −11.17036498275232, −10.13930753141360, −9.735926858147792, −9.175570415261606, −8.625211646209566, −8.065793249441136, −7.352954754420931, −6.651945142688034, −6.220737471876189, −5.326076063291418, −4.612544359363297, −3.895725658864953, −3.306235687675479, −2.365648774826446, −1.854944444729453, −0.5998812622591777, 0.5998812622591777, 1.854944444729453, 2.365648774826446, 3.306235687675479, 3.895725658864953, 4.612544359363297, 5.326076063291418, 6.220737471876189, 6.651945142688034, 7.352954754420931, 8.065793249441136, 8.625211646209566, 9.175570415261606, 9.735926858147792, 10.13930753141360, 11.17036498275232, 11.70331043719405, 11.97007488543712, 12.65482892440077, 13.46000282716391, 14.01621114862507, 14.37789233563134, 14.79935216780154, 15.37174194663751, 16.05710859748804

Graph of the ZZ-function along the critical line