Properties

Label 2-51600-1.1-c1-0-21
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 − 5·7-s + 9-s − 11-s + 3·13-s + 7·19-s + 5·21-s − 4·23-s − 27-s − 3·29-s + 2·31-s + 33-s − 2·37-s − 3·39-s + 8·41-s − 43-s + 7·47-s + 18·49-s + 12·53-s − 7·57-s − 12·59-s + 4·61-s − 5·63-s + 6·67-s + 4·69-s + 8·71-s + 5·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 1.60·19-s + 1.09·21-s − 0.834·23-s − 0.192·27-s − 0.557·29-s + 0.359·31-s + 0.174·33-s − 0.328·37-s − 0.480·39-s + 1.24·41-s − 0.152·43-s + 1.02·47-s + 18/7·49-s + 1.64·53-s − 0.927·57-s − 1.56·59-s + 0.512·61-s − 0.629·63-s + 0.733·67-s + 0.481·69-s + 0.949·71-s + 0.569·77-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

\(L(1)\) \(\approx\) \(1.330971275\)
\(L(\frac12)\) \(\approx\) \(1.330971275\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.32241044983158, −13.78709043106284, −13.48360789172360, −12.89571192018707, −12.47243433940414, −11.99417468871046, −11.46906899796769, −10.82850537017682, −10.29798650202673, −9.863913482374418, −9.377862216745460, −8.949248644875043, −8.164581786791973, −7.399195645410022, −7.126173927774558, −6.340449361175529, −5.959633371493795, −5.577205629823569, −4.825708469904244, −3.828503397439338, −3.662634679811759, −2.887373212499832, −2.226603354034698, −1.080084910940938, −0.4890608776525276, 0.4890608776525276, 1.080084910940938, 2.226603354034698, 2.887373212499832, 3.662634679811759, 3.828503397439338, 4.825708469904244, 5.577205629823569, 5.959633371493795, 6.340449361175529, 7.126173927774558, 7.399195645410022, 8.164581786791973, 8.949248644875043, 9.377862216745460, 9.863913482374418, 10.29798650202673, 10.82850537017682, 11.46906899796769, 11.99417468871046, 12.47243433940414, 12.89571192018707, 13.48360789172360, 13.78709043106284, 14.32241044983158

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