↑

Properties

Label 2-4080-1.1-c1-0-42
Degree 22
Conductor 40804080
Sign −1-1
Analytic cond. 32.578932.5789
Root an. cond. 5.707795.70779
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 − 5-s + 9-s + 2·11-s + 2·13-s + 15-s + 17-s − 8·19-s − 4·23-s + 25-s − 27-s + 2·29-s − 2·31-s − 2·33-s + 6·37-s − 2·39-s + 2·41-s + 2·43-s − 45-s − 6·47-s − 7·49-s − 51-s − 6·53-s − 2·55-s + 8·57-s + 8·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.304·43-s − 0.149·45-s − 0.875·47-s − 49-s − 0.140·51-s − 0.824·53-s − 0.269·55-s + 1.05·57-s + 1.04·59-s + 1.28·61-s + ⋯

Functional equation

Λ(s)=(4080s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4080s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40804080    =    24⋅3⋅5⋅172^{4} \cdot 3 \cdot 5 \cdot 17
Sign: −1-1
Analytic conductor: 32.578932.5789
Root analytic conductor: 5.707795.70779
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4080, ( :1/2), −1)(2,\ 4080,\ (\ :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
5 1+T 1 + T
17 1−T 1 - T
good7 1+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+2T+pT2 1 + 2 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

−8.163998021689167068974326747871, −7.27646840110275465931346359787, −6.38453862032052177963248868561, −6.11004251256253899935993553219, −5.00411018051121736292491828777, −4.20606174033542139955313212186, −3.66694881995610825962971601700, −2.38782684475517298797177638809, −1.29584629424822903147689903793, 0, 1.29584629424822903147689903793, 2.38782684475517298797177638809, 3.66694881995610825962971601700, 4.20606174033542139955313212186, 5.00411018051121736292491828777, 6.11004251256253899935993553219, 6.38453862032052177963248868561, 7.27646840110275465931346359787, 8.163998021689167068974326747871

Graph of the ZZ-function along the critical line