↑

Properties

Label 2-1872-1.1-c1-0-22
Degree 22
Conductor 18721872
Sign −1-1
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s − 2·11-s − 13-s − 6·17-s + 6·19-s + 8·23-s − 25-s − 2·29-s − 10·31-s − 4·35-s − 6·37-s + 6·41-s − 4·43-s − 2·47-s − 3·49-s − 6·53-s + 4·55-s − 10·59-s − 2·61-s + 2·65-s − 10·67-s + 10·71-s + 2·73-s − 4·77-s + 4·79-s − 6·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 0.603·11-s − 0.277·13-s − 1.45·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s − 1.79·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.291·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s − 1.30·59-s − 0.256·61-s + 0.248·65-s − 1.22·67-s + 1.18·71-s + 0.234·73-s − 0.455·77-s + 0.450·79-s − 0.658·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1872, ( :1/2), −1)(2,\ 1872,\ (\ :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
13 1+T 1 + T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−2T+pT2 1 - 2 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 1−6T+pT2 1 - 6 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−8.903536245670668621130507435375, −7.83308528800069718366669266159, −7.49854626403200308211384803397, −6.64015478034033294776010370075, −5.31166823001739669969266033554, −4.85834450825080168955307576161, −3.82761340677172253473232515863, −2.88210740590618426101881210417, −1.61933703544595924973302121161, 0, 1.61933703544595924973302121161, 2.88210740590618426101881210417, 3.82761340677172253473232515863, 4.85834450825080168955307576161, 5.31166823001739669969266033554, 6.64015478034033294776010370075, 7.49854626403200308211384803397, 7.83308528800069718366669266159, 8.903536245670668621130507435375

Graph of the ZZ-function along the critical line