↑

Properties

Label 2-5220-1.1-c1-0-9
Degree 22
Conductor 52205220
Sign 11
Analytic cond. 41.681941.6819
Root an. cond. 6.456156.45615
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 + 4·11-s − 6·13-s + 4·17-s + 4·19-s − 6·23-s + 25-s + 29-s + 2·35-s − 8·37-s + 2·41-s + 4·43-s + 4·47-s − 3·49-s + 2·53-s − 4·55-s − 8·59-s + 10·61-s + 6·65-s − 10·67-s + 8·71-s − 8·77-s + 8·79-s + 6·83-s − 4·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 1.20·11-s − 1.66·13-s + 0.970·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.185·29-s + 0.338·35-s − 1.31·37-s + 0.312·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.28·61-s + 0.744·65-s − 1.22·67-s + 0.949·71-s − 0.911·77-s + 0.900·79-s + 0.658·83-s − 0.433·85-s − 0.635·89-s + ⋯

Functional equation

Λ(s)=(5220s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5220s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52205220    =    22⋅32⋅5⋅292^{2} \cdot 3^{2} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 41.681941.6819
Root analytic conductor: 6.456156.45615
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5220, ( :1/2), 1)(2,\ 5220,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3732608071.373260807
L(12)L(\frac12) ≈\approx 1.3732608071.373260807
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
29 1−T 1 - T
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+8T+pT2 1 + 8 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 1−4T+pT2 1 - 4 T + p T^{2}
53 1−2T+pT2 1 - 2 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+10T+pT2 1 + 10 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1−8T+pT2 1 - 8 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+12T+pT2 1 + 12 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

−8.088083699710393620705951236530, −7.40376064054788239128125112465, −6.91608044558078071309301932539, −6.07032600658004455460921957090, −5.32258447566160792398213499452, −4.44719121575233074558454918551, −3.65226700481618669934198159132, −3.00045546092868765699835197250, −1.89856523418779091680005516342, −0.62579465272769471064720293739, 0.62579465272769471064720293739, 1.89856523418779091680005516342, 3.00045546092868765699835197250, 3.65226700481618669934198159132, 4.44719121575233074558454918551, 5.32258447566160792398213499452, 6.07032600658004455460921957090, 6.91608044558078071309301932539, 7.40376064054788239128125112465, 8.088083699710393620705951236530

Graph of the ZZ-function along the critical line