↑

Properties

Label 2-3960-1.1-c1-0-13
Degree 22
Conductor 39603960
Sign 11
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
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  + 5-s − 2·7-s + 11-s − 4·13-s + 4·17-s + 8·19-s − 4·23-s + 25-s − 2·29-s + 8·31-s − 2·35-s − 2·37-s + 2·41-s − 6·43-s − 12·47-s − 3·49-s − 6·53-s + 55-s + 12·59-s + 10·61-s − 4·65-s + 16·67-s + 4·73-s − 2·77-s + 12·79-s + 6·83-s + 4·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.301·11-s − 1.10·13-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s − 0.914·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.134·55-s + 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.95·67-s + 0.468·73-s − 0.227·77-s + 1.35·79-s + 0.658·83-s + 0.433·85-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), 1)(2,\ 3960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8487081101.848708110
L(12)L(\frac12) ≈\approx 1.8487081101.848708110
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
5 1−T 1 - T
11 1−T 1 - T
good7 1+2T+pT2 1 + 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−4T+pT2 1 - 4 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 1−8T+pT2 1 - 8 T + 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+12T+pT2 1 + 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−12T+pT2 1 - 12 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 1−14T+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

−8.352705169495267588684878045501, −7.74466140787582305697515866698, −6.91334750979845635259915312078, −6.32926538480038172431213566680, −5.39425694270498179229785079753, −4.91038842310775196411268398885, −3.63649248518632718778936997907, −3.05297823603964657597649779958, −2.01555607049808295279860849648, −0.78205375494121749325567483276, 0.78205375494121749325567483276, 2.01555607049808295279860849648, 3.05297823603964657597649779958, 3.63649248518632718778936997907, 4.91038842310775196411268398885, 5.39425694270498179229785079753, 6.32926538480038172431213566680, 6.91334750979845635259915312078, 7.74466140787582305697515866698, 8.352705169495267588684878045501

Graph of the ZZ-function along the critical line