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Properties

Label 2-320892-1.1-c1-0-19
Degree 22
Conductor 320892320892
Sign −1-1
Analytic cond. 2562.332562.33
Root an. cond. 50.619550.6195
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-s + 2·5-s + 9-s − 13-s − 2·15-s − 17-s + 4·23-s − 25-s − 27-s − 6·29-s + 4·31-s − 8·37-s + 39-s + 10·43-s + 2·45-s + 8·47-s − 7·49-s + 51-s + 6·53-s − 12·59-s + 12·61-s − 2·65-s + 6·67-s − 4·69-s + 14·73-s + 75-s + 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.277·13-s − 0.516·15-s − 0.242·17-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 1.31·37-s + 0.160·39-s + 1.52·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.140·51-s + 0.824·53-s − 1.56·59-s + 1.53·61-s − 0.248·65-s + 0.733·67-s − 0.481·69-s + 1.63·73-s + 0.115·75-s + 0.900·79-s + ⋯

Functional equation

Λ(s)=(320892s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(320892s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320892320892    =    22⋅3⋅112⋅13⋅172^{2} \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17
Sign: −1-1
Analytic conductor: 2562.332562.33
Root analytic conductor: 50.619550.6195
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 320892, ( :1/2), −1)(2,\ 320892,\ (\ :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+T 1 + T
11 1 1
13 1+T 1 + T
17 1+T 1 + T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1−6T+pT2 1 - 6 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−12.79430587975967, −12.35108253800161, −12.07775666310462, −11.42243421372574, −10.86645104514251, −10.75326031281568, −10.13280541636895, −9.638161264935266, −9.201544990235653, −9.016755132716933, −8.092300988023596, −7.850942389184672, −7.135708996198313, −6.679194191138780, −6.422077166461182, −5.602502932033179, −5.453706176871779, −5.004010048094508, −4.279329113114363, −3.846927790055289, −3.196108059393634, −2.407988266741773, −2.141734378723537, −1.382425427325754, −0.8070682625520731, 0, 0.8070682625520731, 1.382425427325754, 2.141734378723537, 2.407988266741773, 3.196108059393634, 3.846927790055289, 4.279329113114363, 5.004010048094508, 5.453706176871779, 5.602502932033179, 6.422077166461182, 6.679194191138780, 7.135708996198313, 7.850942389184672, 8.092300988023596, 9.016755132716933, 9.201544990235653, 9.638161264935266, 10.13280541636895, 10.75326031281568, 10.86645104514251, 11.42243421372574, 12.07775666310462, 12.35108253800161, 12.79430587975967

Graph of the ZZ-function along the critical line