Properties

Label 2-51600-1.1-c1-0-20
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 − 4·7-s + 9-s + 4·11-s − 4·13-s − 4·17-s − 4·19-s + 4·21-s + 8·23-s − 27-s + 6·29-s + 4·31-s − 4·33-s − 2·37-s + 4·39-s + 10·41-s − 43-s + 4·47-s + 9·49-s + 4·51-s + 2·53-s + 4·57-s + 12·59-s − 4·63-s + 8·67-s − 8·69-s + 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.970·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.152·43-s + 0.583·47-s + 9/7·49-s + 0.560·51-s + 0.274·53-s + 0.529·57-s + 1.56·59-s − 0.503·63-s + 0.977·67-s − 0.963·69-s + 1.42·71-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

\(L(1)\) \(\approx\) \(1.356982529\)
\(L(\frac12)\) \(\approx\) \(1.356982529\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.57054136092293, −13.93288372952460, −13.33705976969554, −12.82809144832201, −12.46122807877989, −12.08578135012954, −11.35685138291936, −10.95190700773856, −10.31434702843558, −9.795555698932940, −9.357712839189096, −8.903604434430710, −8.331590205018283, −7.356632650329248, −6.848211557769570, −6.548492768756733, −6.201389044783886, −5.289644030534003, −4.769745522857566, −4.084660855438559, −3.628952870839591, −2.638958701078731, −2.373610032129625, −1.087568689194519, −0.4887947982332246, 0.4887947982332246, 1.087568689194519, 2.373610032129625, 2.638958701078731, 3.628952870839591, 4.084660855438559, 4.769745522857566, 5.289644030534003, 6.201389044783886, 6.548492768756733, 6.848211557769570, 7.356632650329248, 8.331590205018283, 8.903604434430710, 9.357712839189096, 9.795555698932940, 10.31434702843558, 10.95190700773856, 11.35685138291936, 12.08578135012954, 12.46122807877989, 12.82809144832201, 13.33705976969554, 13.93288372952460, 14.57054136092293

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