↑

Properties

Label 2-3192-1.1-c1-0-53
Degree 22
Conductor 31923192
Sign −1-1
Analytic cond. 25.488225.4882
Root an. cond. 5.048585.04858
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 2·13-s − 8·17-s − 19-s + 21-s − 4·23-s − 5·25-s + 27-s − 2·37-s − 2·39-s + 2·41-s − 4·43-s − 6·47-s + 49-s − 8·51-s − 12·53-s − 57-s − 4·59-s − 2·61-s + 63-s + 4·67-s − 4·69-s + 6·71-s + 6·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.94·17-s − 0.229·19-s + 0.218·21-s − 0.834·23-s − 25-s + 0.192·27-s − 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s − 1.12·51-s − 1.64·53-s − 0.132·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + 0.488·67-s − 0.481·69-s + 0.712·71-s + 0.702·73-s − 0.577·75-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 25.488225.4882
Root analytic conductor: 5.048585.04858
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3192, ( :1/2), −1)(2,\ 3192,\ (\ :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−T 1 - T
7 1−T 1 - T
19 1+T 1 + T
good5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+8T+pT2 1 + 8 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 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+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−6T+pT2 1 - 6 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 1−18T+pT2 1 - 18 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−8.208329763915009155875693274441, −7.75381208178629701986032179519, −6.78432954817853412246616257104, −6.20105211219491075074899961625, −5.02692945253514422792587878059, −4.40011931011779862680526321493, −3.55875172199588887914229143879, −2.39741119584776535278731996108, −1.79152280881413347668477476302, 0, 1.79152280881413347668477476302, 2.39741119584776535278731996108, 3.55875172199588887914229143879, 4.40011931011779862680526321493, 5.02692945253514422792587878059, 6.20105211219491075074899961625, 6.78432954817853412246616257104, 7.75381208178629701986032179519, 8.208329763915009155875693274441

Graph of the ZZ-function along the critical line