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Properties

Label 2-64800-1.1-c1-0-2
Degree 22
Conductor 6480064800
Sign 11
Analytic cond. 517.430517.430
Root an. cond. 22.747122.7471
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 11-s + 17-s − 7·19-s + 6·23-s − 2·29-s + 10·31-s − 8·37-s + 5·41-s + 5·43-s − 4·47-s + 9·49-s + 10·53-s + 9·59-s − 10·61-s + 3·67-s − 6·71-s + 11·73-s + 4·77-s + 10·79-s + 4·83-s − 18·89-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.301·11-s + 0.242·17-s − 1.60·19-s + 1.25·23-s − 0.371·29-s + 1.79·31-s − 1.31·37-s + 0.780·41-s + 0.762·43-s − 0.583·47-s + 9/7·49-s + 1.37·53-s + 1.17·59-s − 1.28·61-s + 0.366·67-s − 0.712·71-s + 1.28·73-s + 0.455·77-s + 1.12·79-s + 0.439·83-s − 1.90·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

Λ(s)=(64800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(64800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6480064800    =    25⋅34⋅522^{5} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 517.430517.430
Root analytic conductor: 22.747122.7471
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 64800, ( :1/2), 1)(2,\ 64800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1889750471.188975047
L(12)L(\frac12) ≈\approx 1.1889750471.188975047
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 1
good7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−T+pT2 1 - T + p T^{2}
19 1+7T+pT2 1 + 7 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 1−10T+pT2 1 - 10 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−9T+pT2 1 - 9 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+18T+pT2 1 + 18 T + p T^{2}
97 1+7T+pT2 1 + 7 T + p T^{2}
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   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

−14.10737267574075, −13.64420977256419, −13.16985457612533, −12.66822897783568, −12.44815768013055, −11.79691686508104, −11.07185755642010, −10.63694722993403, −10.11659993374844, −9.748919756132755, −9.027674778125814, −8.722091720483545, −8.093731143522155, −7.381374802578964, −6.805366254380570, −6.454214127161482, −5.928521655024046, −5.258937763386394, −4.613064197984708, −3.929085784768980, −3.417807700168452, −2.680897804435617, −2.330261122592645, −1.193220861211701, −0.3914088102414002, 0.3914088102414002, 1.193220861211701, 2.330261122592645, 2.680897804435617, 3.417807700168452, 3.929085784768980, 4.613064197984708, 5.258937763386394, 5.928521655024046, 6.454214127161482, 6.805366254380570, 7.381374802578964, 8.093731143522155, 8.722091720483545, 9.027674778125814, 9.748919756132755, 10.11659993374844, 10.63694722993403, 11.07185755642010, 11.79691686508104, 12.44815768013055, 12.66822897783568, 13.16985457612533, 13.64420977256419, 14.10737267574075

Graph of the ZZ-function along the critical line