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Properties

Label 2-100800-1.1-c1-0-82
Degree 22
Conductor 100800100800
Sign 11
Analytic cond. 804.892804.892
Root an. cond. 28.370628.3706
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 3·11-s + 13-s + 7·17-s − 6·23-s − 5·29-s − 2·31-s + 2·37-s − 2·41-s + 4·43-s + 3·47-s + 49-s − 6·53-s − 10·59-s + 8·61-s − 2·67-s − 8·71-s − 6·73-s − 3·77-s + 5·79-s − 4·83-s − 91-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.904·11-s + 0.277·13-s + 1.69·17-s − 1.25·23-s − 0.928·29-s − 0.359·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s − 1.30·59-s + 1.02·61-s − 0.244·67-s − 0.949·71-s − 0.702·73-s − 0.341·77-s + 0.562·79-s − 0.439·83-s − 0.104·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

Λ(s)=(100800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(100800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 100800100800    =    26⋅32⋅52⋅72^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 804.892804.892
Root analytic conductor: 28.370628.3706
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 100800, ( :1/2), 1)(2,\ 100800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1429120472.142912047
L(12)L(\frac12) ≈\approx 2.1429120472.142912047
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
7 1+T 1 + T
good11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−7T+pT2 1 - 7 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+7T+pT2 1 + 7 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

−13.85716991223592, −13.21448823444661, −12.76332911593133, −12.13709368555031, −11.96823634818984, −11.36897324952286, −10.75617064473575, −10.29781394305843, −9.652965079693843, −9.464253251852160, −8.830132382473875, −8.231923685080304, −7.654490600934044, −7.358520663018216, −6.553669920442149, −6.134848716030353, −5.635056486927186, −5.172909763127934, −4.162591689426138, −3.969730968036363, −3.279210365047187, −2.757485012635521, −1.783293149723287, −1.360462621967082, −0.4699492985667051, 0.4699492985667051, 1.360462621967082, 1.783293149723287, 2.757485012635521, 3.279210365047187, 3.969730968036363, 4.162591689426138, 5.172909763127934, 5.635056486927186, 6.134848716030353, 6.553669920442149, 7.358520663018216, 7.654490600934044, 8.231923685080304, 8.830132382473875, 9.464253251852160, 9.652965079693843, 10.29781394305843, 10.75617064473575, 11.36897324952286, 11.96823634818984, 12.13709368555031, 12.76332911593133, 13.21448823444661, 13.85716991223592

Graph of the ZZ-function along the critical line