Properties

Label 2-51600-1.1-c1-0-44
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 + 4·11-s − 2·13-s + 2·17-s + 4·19-s − 2·21-s + 2·23-s + 27-s + 10·29-s + 4·31-s + 4·33-s + 8·37-s − 2·39-s + 6·41-s + 43-s + 2·47-s − 3·49-s + 2·51-s + 12·53-s + 4·57-s − 4·59-s − 8·61-s − 2·63-s + 4·67-s + 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.436·21-s + 0.417·23-s + 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.696·33-s + 1.31·37-s − 0.320·39-s + 0.937·41-s + 0.152·43-s + 0.291·47-s − 3/7·49-s + 0.280·51-s + 1.64·53-s + 0.529·57-s − 0.520·59-s − 1.02·61-s − 0.251·63-s + 0.488·67-s + 0.240·69-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

\(L(1)\) \(\approx\) \(3.860952363\)
\(L(\frac12)\) \(\approx\) \(3.860952363\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.40759459587566, −14.02440436193388, −13.55815188341475, −13.01773547028685, −12.32818991095906, −12.05039146117819, −11.58711185439465, −10.79182373364102, −10.20936462231194, −9.701385923847409, −9.341656670186271, −8.886970858467201, −8.168602603850165, −7.670971610005652, −7.062660010089942, −6.543917802782275, −6.084411428290958, −5.346301935989330, −4.531020677527680, −4.169432540491237, −3.301041897332629, −2.936174327879450, −2.287186559928527, −1.203155564991172, −0.7659675056567394, 0.7659675056567394, 1.203155564991172, 2.287186559928527, 2.936174327879450, 3.301041897332629, 4.169432540491237, 4.531020677527680, 5.346301935989330, 6.084411428290958, 6.543917802782275, 7.062660010089942, 7.670971610005652, 8.168602603850165, 8.886970858467201, 9.341656670186271, 9.701385923847409, 10.20936462231194, 10.79182373364102, 11.58711185439465, 12.05039146117819, 12.32818991095906, 13.01773547028685, 13.55815188341475, 14.02440436193388, 14.40759459587566

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