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Properties

Label 2-171600-1.1-c1-0-119
Degree 22
Conductor 171600171600
Sign −1-1
Analytic cond. 1370.231370.23
Root an. cond. 37.016637.0166
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 − 3·7-s + 9-s + 11-s + 13-s − 2·17-s − 5·19-s − 3·21-s − 3·23-s + 27-s − 2·29-s + 6·31-s + 33-s − 4·37-s + 39-s + 2·41-s − 6·43-s − 10·47-s + 2·49-s − 2·51-s − 9·53-s − 5·57-s + 6·59-s − 6·61-s − 3·63-s + 16·67-s − 3·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.485·17-s − 1.14·19-s − 0.654·21-s − 0.625·23-s + 0.192·27-s − 0.371·29-s + 1.07·31-s + 0.174·33-s − 0.657·37-s + 0.160·39-s + 0.312·41-s − 0.914·43-s − 1.45·47-s + 2/7·49-s − 0.280·51-s − 1.23·53-s − 0.662·57-s + 0.781·59-s − 0.768·61-s − 0.377·63-s + 1.95·67-s − 0.361·69-s + ⋯

Functional equation

Λ(s)=(171600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(171600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171600171600    =    24⋅3⋅52⋅11⋅132^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13
Sign: −1-1
Analytic conductor: 1370.231370.23
Root analytic conductor: 37.016637.0166
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 171600, ( :1/2), −1)(2,\ 171600,\ (\ :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
5 1 1
11 1−T 1 - T
13 1−T 1 - T
good7 1+3T+pT2 1 + 3 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+12T+pT2 1 + 12 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+5T+pT2 1 + 5 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−4T+pT2 1 - 4 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

−13.37383276318104, −12.92585178539615, −12.73203767722253, −12.11531460543660, −11.54945744969002, −11.10336758566822, −10.46329690006933, −10.05927034517149, −9.601480029396082, −9.243963006334319, −8.547319352667659, −8.313474733811055, −7.784346190069319, −6.952685462505953, −6.681186120029474, −6.247086024007652, −5.777534332508944, −4.888234563267209, −4.468198319132489, −3.846293661688260, −3.329513312919490, −2.955489687117194, −2.067611695705863, −1.795772943947896, −0.7292363782915662, 0, 0.7292363782915662, 1.795772943947896, 2.067611695705863, 2.955489687117194, 3.329513312919490, 3.846293661688260, 4.468198319132489, 4.888234563267209, 5.777534332508944, 6.247086024007652, 6.681186120029474, 6.952685462505953, 7.784346190069319, 8.313474733811055, 8.547319352667659, 9.243963006334319, 9.601480029396082, 10.05927034517149, 10.46329690006933, 11.10336758566822, 11.54945744969002, 12.11531460543660, 12.73203767722253, 12.92585178539615, 13.37383276318104

Graph of the ZZ-function along the critical line