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Properties

Label 2-64800-1.1-c1-0-33
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  − 2·7-s + 2·11-s − 13-s − 3·17-s + 2·19-s − 6·23-s + 29-s − 8·31-s − 37-s − 2·41-s − 10·43-s − 4·47-s − 3·49-s + 10·53-s + 4·59-s + 9·61-s + 14·67-s + 10·71-s + 9·73-s − 4·77-s + 10·79-s − 12·83-s + 11·89-s + 2·91-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.727·17-s + 0.458·19-s − 1.25·23-s + 0.185·29-s − 1.43·31-s − 0.164·37-s − 0.312·41-s − 1.52·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s + 0.520·59-s + 1.15·61-s + 1.71·67-s + 1.18·71-s + 1.05·73-s − 0.455·77-s + 1.12·79-s − 1.31·83-s + 1.16·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-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+2T+pT2 1 + 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+10T+pT2 1 + 10 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 1−4T+pT2 1 - 4 T + p T^{2}
61 1−9T+pT2 1 - 9 T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−11T+pT2 1 - 11 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−14.54542356614367, −13.89984477235497, −13.51018058354977, −12.93969616977407, −12.51198371228182, −11.97282793115552, −11.44688906806942, −11.06858578806835, −10.26634659781327, −9.838060863280513, −9.514883965516438, −8.848455525140596, −8.348767585557635, −7.811964786672133, −6.992737751257836, −6.732864844853123, −6.222670597474059, −5.436359086073970, −5.070625898676798, −4.187956893033775, −3.665166984068944, −3.268708569989944, −2.233580308995340, −1.905200908253443, −0.8078974052523228, 0, 0.8078974052523228, 1.905200908253443, 2.233580308995340, 3.268708569989944, 3.665166984068944, 4.187956893033775, 5.070625898676798, 5.436359086073970, 6.222670597474059, 6.732864844853123, 6.992737751257836, 7.811964786672133, 8.348767585557635, 8.848455525140596, 9.514883965516438, 9.838060863280513, 10.26634659781327, 11.06858578806835, 11.44688906806942, 11.97282793115552, 12.51198371228182, 12.93969616977407, 13.51018058354977, 13.89984477235497, 14.54542356614367

Graph of the ZZ-function along the critical line