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Properties

Label 2-64800-1.1-c1-0-51
Degree 22
Conductor 6480064800
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 3·11-s + 2·13-s − 6·17-s − 5·19-s + 8·23-s + 3·29-s − 5·31-s + 4·37-s − 9·41-s − 4·43-s − 2·47-s + 9·49-s + 3·59-s + 6·61-s + 6·67-s + 5·71-s − 12·77-s − 8·79-s + 6·83-s − 89-s + 8·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.904·11-s + 0.554·13-s − 1.45·17-s − 1.14·19-s + 1.66·23-s + 0.557·29-s − 0.898·31-s + 0.657·37-s − 1.40·41-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.390·59-s + 0.768·61-s + 0.733·67-s + 0.593·71-s − 1.36·77-s − 0.900·79-s + 0.658·83-s − 0.105·89-s + 0.838·91-s + 1.01·97-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: −1-1
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: 11
Selberg data: (2, 64800, ( :1/2), −1)(2,\ 64800,\ (\ :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
5 1 1
good7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−6T+pT2 1 - 6 T + p T^{2}
71 1−5T+pT2 1 - 5 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+T+pT2 1 + T + p T^{2}
97 1−10T+pT2 1 - 10 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

−14.62941757212447, −13.96169231479799, −13.36006944183295, −13.05854295475268, −12.64249815691916, −11.77134332053173, −11.32606562109564, −10.96082030013222, −10.64651176786219, −10.04150446694792, −9.176734841943468, −8.743756765019292, −8.263519269360284, −8.013781180513759, −7.066243726300395, −6.833674400280805, −6.098737350375963, −5.276081293551587, −4.979549523016532, −4.461890372449145, −3.837914432512838, −2.986975914610972, −2.266340578320396, −1.815477301754546, −0.9939884782721783, 0, 0.9939884782721783, 1.815477301754546, 2.266340578320396, 2.986975914610972, 3.837914432512838, 4.461890372449145, 4.979549523016532, 5.276081293551587, 6.098737350375963, 6.833674400280805, 7.066243726300395, 8.013781180513759, 8.263519269360284, 8.743756765019292, 9.176734841943468, 10.04150446694792, 10.64651176786219, 10.96082030013222, 11.32606562109564, 11.77134332053173, 12.64249815691916, 13.05854295475268, 13.36006944183295, 13.96169231479799, 14.62941757212447

Graph of the ZZ-function along the critical line