Properties

Label 2-51600-1.1-c1-0-32
Degree $2$
Conductor $51600$
Sign $1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s + 9-s + 3·11-s − 5·13-s − 5·17-s + 2·21-s + 27-s − 2·29-s + 8·31-s + 3·33-s + 4·37-s − 5·39-s + 10·41-s + 43-s − 3·47-s − 3·49-s − 5·51-s − 2·53-s − 9·59-s − 4·61-s + 2·63-s + 13·67-s + 6·71-s − 8·73-s + 6·77-s + 5·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.904·11-s − 1.38·13-s − 1.21·17-s + 0.436·21-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.522·33-s + 0.657·37-s − 0.800·39-s + 1.56·41-s + 0.152·43-s − 0.437·47-s − 3/7·49-s − 0.700·51-s − 0.274·53-s − 1.17·59-s − 0.512·61-s + 0.251·63-s + 1.58·67-s + 0.712·71-s − 0.936·73-s + 0.683·77-s + 0.562·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.137883116\)
\(L(\frac12)\) \(\approx\) \(3.137883116\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
43 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.35073320452132, −14.18046090678456, −13.59102018218276, −12.91076493928445, −12.50362544334486, −11.91737823663184, −11.37615591573045, −11.01719976917835, −10.29001460372594, −9.668775549366504, −9.290708119978733, −8.835556730971857, −8.051245871883114, −7.814919782442171, −7.118944047595196, −6.572197478039275, −6.062060421879835, −5.154387956937845, −4.505388584378325, −4.374796172764042, −3.469828443975788, −2.653040956108134, −2.211737690698459, −1.495312428222691, −0.5964078377356274, 0.5964078377356274, 1.495312428222691, 2.211737690698459, 2.653040956108134, 3.469828443975788, 4.374796172764042, 4.505388584378325, 5.154387956937845, 6.062060421879835, 6.572197478039275, 7.118944047595196, 7.814919782442171, 8.051245871883114, 8.835556730971857, 9.290708119978733, 9.668775549366504, 10.29001460372594, 11.01719976917835, 11.37615591573045, 11.91737823663184, 12.50362544334486, 12.91076493928445, 13.59102018218276, 14.18046090678456, 14.35073320452132

Graph of the $Z$-function along the critical line