Properties

Label 2-51600-1.1-c1-0-45
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 + 13-s + 2·17-s + 8·19-s − 2·21-s + 3·23-s − 27-s + 8·29-s − 9·31-s − 3·33-s + 10·37-s − 39-s − 3·41-s + 43-s − 7·47-s − 3·49-s − 2·51-s + 5·53-s − 8·57-s + 9·59-s + 2·61-s + 2·63-s − 8·67-s − 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.485·17-s + 1.83·19-s − 0.436·21-s + 0.625·23-s − 0.192·27-s + 1.48·29-s − 1.61·31-s − 0.522·33-s + 1.64·37-s − 0.160·39-s − 0.468·41-s + 0.152·43-s − 1.02·47-s − 3/7·49-s − 0.280·51-s + 0.686·53-s − 1.05·57-s + 1.17·59-s + 0.256·61-s + 0.251·63-s − 0.977·67-s − 0.361·69-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

\(L(1)\) \(\approx\) \(3.187600337\)
\(L(\frac12)\) \(\approx\) \(3.187600337\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.54757360244970, −14.02808194177407, −13.48281702640973, −12.94813408494133, −12.28075284914447, −11.85026830708244, −11.39155377786929, −11.11533066525806, −10.39486049427333, −9.777343977577921, −9.400966399061466, −8.765987996467928, −8.142934862778672, −7.580970204039859, −7.097370208790405, −6.483595227516096, −5.906995171543325, −5.225946287230390, −4.930436995238092, −4.164792672980344, −3.520595596177711, −2.926193766140677, −1.919222485111997, −1.192205976769917, −0.7696137595730807, 0.7696137595730807, 1.192205976769917, 1.919222485111997, 2.926193766140677, 3.520595596177711, 4.164792672980344, 4.930436995238092, 5.225946287230390, 5.906995171543325, 6.483595227516096, 7.097370208790405, 7.580970204039859, 8.142934862778672, 8.765987996467928, 9.400966399061466, 9.777343977577921, 10.39486049427333, 11.11533066525806, 11.39155377786929, 11.85026830708244, 12.28075284914447, 12.94813408494133, 13.48281702640973, 14.02808194177407, 14.54757360244970

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