↑

Properties

Label 2-3168-1.1-c1-0-33
Degree 22
Conductor 31683168
Sign −1-1
Analytic cond. 25.296625.2966
Root an. cond. 5.029575.02957
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  − 2·5-s + 11-s + 2·17-s − 6·19-s + 6·23-s − 25-s + 2·29-s − 2·37-s − 2·41-s − 6·43-s + 6·47-s − 7·49-s + 2·53-s − 2·55-s + 8·61-s − 12·67-s − 6·71-s + 6·73-s − 12·79-s − 12·83-s − 4·85-s − 8·89-s + 12·95-s + 6·97-s − 2·101-s − 12·107-s + 4·113-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.301·11-s + 0.485·17-s − 1.37·19-s + 1.25·23-s − 1/5·25-s + 0.371·29-s − 0.328·37-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 49-s + 0.274·53-s − 0.269·55-s + 1.02·61-s − 1.46·67-s − 0.712·71-s + 0.702·73-s − 1.35·79-s − 1.31·83-s − 0.433·85-s − 0.847·89-s + 1.23·95-s + 0.609·97-s − 0.199·101-s − 1.16·107-s + 0.376·113-s + ⋯

Functional equation

Λ(s)=(3168s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3168s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31683168    =    25⋅32⋅112^{5} \cdot 3^{2} \cdot 11
Sign: −1-1
Analytic conductor: 25.296625.2966
Root analytic conductor: 5.029575.02957
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3168, ( :1/2), −1)(2,\ 3168,\ (\ :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
11 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−2T+pT2 1 - 2 T + 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+6T+pT2 1 + 6 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+12T+pT2 1 + 12 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+12T+pT2 1 + 12 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+8T+pT2 1 + 8 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−8.422830637318465495529175571444, −7.52836968068358160938962254123, −6.90231048077484383576044392193, −6.11937289864350764583968354951, −5.12891869990477172704763936246, −4.32434755255141440842877247524, −3.61400534899424541377714722299, −2.68380740919008903645509592858, −1.39742958097402412766819249463, 0, 1.39742958097402412766819249463, 2.68380740919008903645509592858, 3.61400534899424541377714722299, 4.32434755255141440842877247524, 5.12891869990477172704763936246, 6.11937289864350764583968354951, 6.90231048077484383576044392193, 7.52836968068358160938962254123, 8.422830637318465495529175571444

Graph of the ZZ-function along the critical line