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Properties

Label 2-103428-1.1-c1-0-27
Degree 22
Conductor 103428103428
Sign −1-1
Analytic cond. 825.876825.876
Root an. cond. 28.738028.7380
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  + 3·5-s + 4·7-s − 11-s − 17-s + 4·19-s − 3·23-s + 4·25-s − 3·29-s + 12·35-s − 2·37-s + 2·41-s − 43-s − 4·47-s + 9·49-s − 3·55-s + 2·59-s − 8·61-s − 5·67-s − 4·77-s − 8·79-s − 3·85-s + 6·89-s + 12·95-s − 12·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.51·7-s − 0.301·11-s − 0.242·17-s + 0.917·19-s − 0.625·23-s + 4/5·25-s − 0.557·29-s + 2.02·35-s − 0.328·37-s + 0.312·41-s − 0.152·43-s − 0.583·47-s + 9/7·49-s − 0.404·55-s + 0.260·59-s − 1.02·61-s − 0.610·67-s − 0.455·77-s − 0.900·79-s − 0.325·85-s + 0.635·89-s + 1.23·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

Λ(s)=(103428s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(103428s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 103428103428    =    22⋅32⋅132⋅172^{2} \cdot 3^{2} \cdot 13^{2} \cdot 17
Sign: −1-1
Analytic conductor: 825.876825.876
Root analytic conductor: 28.738028.7380
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 103428, ( :1/2), −1)(2,\ 103428,\ (\ :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
13 1 1
17 1+T 1 + T
good5 1−3T+pT2 1 - 3 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−2T+pT2 1 - 2 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+pT2 1 + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−13.83319128484446, −13.67378613238427, −13.14977116868020, −12.48252602699379, −12.01438684738302, −11.39558831695254, −11.11306039965104, −10.42480636154247, −10.14453927961702, −9.429533144119777, −9.181526893113834, −8.432508944414825, −8.044371832707103, −7.491930214030319, −6.985194427871059, −6.233336907462283, −5.763257161885355, −5.293717916477636, −4.890121580918220, −4.281738165652890, −3.559717142301885, −2.700668059010489, −2.237404026528188, −1.522578248291845, −1.275326291188472, 0, 1.275326291188472, 1.522578248291845, 2.237404026528188, 2.700668059010489, 3.559717142301885, 4.281738165652890, 4.890121580918220, 5.293717916477636, 5.763257161885355, 6.233336907462283, 6.985194427871059, 7.491930214030319, 8.044371832707103, 8.432508944414825, 9.181526893113834, 9.429533144119777, 10.14453927961702, 10.42480636154247, 11.11306039965104, 11.39558831695254, 12.01438684738302, 12.48252602699379, 13.14977116868020, 13.67378613238427, 13.83319128484446

Graph of the ZZ-function along the critical line