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Properties

Label 2-64800-1.1-c1-0-50
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  + 3·11-s + 4·13-s − 3·17-s − 4·19-s − 2·23-s − 2·29-s + 4·31-s − 4·37-s − 10·41-s + 7·43-s + 4·47-s − 7·49-s + 8·53-s − 59-s + 4·61-s − 12·67-s + 14·71-s + 2·73-s + 83-s + 89-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.904·11-s + 1.10·13-s − 0.727·17-s − 0.917·19-s − 0.417·23-s − 0.371·29-s + 0.718·31-s − 0.657·37-s − 1.56·41-s + 1.06·43-s + 0.583·47-s − 49-s + 1.09·53-s − 0.130·59-s + 0.512·61-s − 1.46·67-s + 1.66·71-s + 0.234·73-s + 0.109·83-s + 0.105·89-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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+pT2 1 + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1−7T+pT2 1 - 7 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−8T+pT2 1 - 8 T + p T^{2}
59 1+T+pT2 1 + T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−14T+pT2 1 - 14 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−T+pT2 1 - T + p T^{2}
89 1−T+pT2 1 - T + p T^{2}
97 1+17T+pT2 1 + 17 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

−14.38968875559125, −13.95302066876820, −13.47889799378432, −13.07820115148017, −12.42080235209582, −11.92914161338975, −11.49856687387492, −10.84693498177813, −10.59276850935540, −9.885907035889173, −9.291513665446706, −8.807099177673785, −8.384190202210416, −7.922557182717953, −6.959044910541980, −6.724682467292659, −6.138726561217307, −5.638134053117438, −4.869667803334284, −4.174554189536374, −3.856627361052723, −3.170679105047680, −2.303998793141186, −1.709933638037258, −0.9893344904698217, 0, 0.9893344904698217, 1.709933638037258, 2.303998793141186, 3.170679105047680, 3.856627361052723, 4.174554189536374, 4.869667803334284, 5.638134053117438, 6.138726561217307, 6.724682467292659, 6.959044910541980, 7.922557182717953, 8.384190202210416, 8.807099177673785, 9.291513665446706, 9.885907035889173, 10.59276850935540, 10.84693498177813, 11.49856687387492, 11.92914161338975, 12.42080235209582, 13.07820115148017, 13.47889799378432, 13.95302066876820, 14.38968875559125

Graph of the ZZ-function along the critical line