Properties

Label 2-51600-1.1-c1-0-27
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 + 4·21-s − 4·23-s + 27-s − 6·29-s + 4·31-s − 4·33-s + 2·37-s − 4·39-s + 2·41-s + 43-s − 8·47-s + 9·49-s + 6·53-s + 4·59-s − 4·61-s + 4·63-s + 8·67-s − 4·69-s − 4·71-s + 10·73-s − 16·77-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.872·21-s − 0.834·23-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 + 0.312·41-s + 0.152·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s − 0.512·61-s + 0.503·63-s + 0.977·67-s − 0.481·69-s − 0.474·71-s + 1.17·73-s − 1.82·77-s + 0.900·79-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

\(L(1)\) \(\approx\) \(2.670020576\)
\(L(\frac12)\) \(\approx\) \(2.670020576\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.44905261019418, −14.09142303538956, −13.56577639751325, −12.93810897941474, −12.56561956235535, −11.79881876430775, −11.50935065284133, −10.82241657827307, −10.37473850650101, −9.804899817559059, −9.352091154118724, −8.530745306673847, −8.150708953547150, −7.700524656408863, −7.389654910549342, −6.620830545546995, −5.738605228271687, −5.229641753792626, −4.757718065435974, −4.232932002179435, −3.468154989752984, −2.580799795458357, −2.213117929021005, −1.586645073416971, −0.5298913387586935, 0.5298913387586935, 1.586645073416971, 2.213117929021005, 2.580799795458357, 3.468154989752984, 4.232932002179435, 4.757718065435974, 5.229641753792626, 5.738605228271687, 6.620830545546995, 7.389654910549342, 7.700524656408863, 8.150708953547150, 8.530745306673847, 9.352091154118724, 9.804899817559059, 10.37473850650101, 10.82241657827307, 11.50935065284133, 11.79881876430775, 12.56561956235535, 12.93810897941474, 13.56577639751325, 14.09142303538956, 14.44905261019418

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