Properties

Label 2-51600-1.1-c1-0-101
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s − 11-s + 13-s − 6·17-s − 8·19-s − 2·21-s − 9·23-s + 27-s − 4·29-s − 5·31-s − 33-s − 2·37-s + 39-s − 11·41-s + 43-s − 11·47-s − 3·49-s − 6·51-s − 3·53-s − 8·57-s + 5·59-s − 14·61-s − 2·63-s + 16·67-s − 9·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 1.45·17-s − 1.83·19-s − 0.436·21-s − 1.87·23-s + 0.192·27-s − 0.742·29-s − 0.898·31-s − 0.174·33-s − 0.328·37-s + 0.160·39-s − 1.71·41-s + 0.152·43-s − 1.60·47-s − 3/7·49-s − 0.840·51-s − 0.412·53-s − 1.05·57-s + 0.650·59-s − 1.79·61-s − 0.251·63-s + 1.95·67-s − 1.08·69-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: \(2\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
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

−14.94220123121196, −14.57692214582206, −13.90943038319876, −13.29933280503938, −13.14574947149285, −12.55605456519257, −12.09121939910151, −11.22702672716700, −10.92981927068563, −10.27808762827158, −9.788242889497924, −9.315845811565950, −8.628624243427005, −8.315018337967563, −7.801899602351755, −6.910026548869930, −6.548266885069932, −6.146259160669863, −5.322139282664646, −4.609047390907404, −3.953826429202707, −3.613435596608812, −2.782691235943118, −1.983409428538851, −1.784547743125972, 0, 0, 1.784547743125972, 1.983409428538851, 2.782691235943118, 3.613435596608812, 3.953826429202707, 4.609047390907404, 5.322139282664646, 6.146259160669863, 6.548266885069932, 6.910026548869930, 7.801899602351755, 8.315018337967563, 8.628624243427005, 9.315845811565950, 9.788242889497924, 10.27808762827158, 10.92981927068563, 11.22702672716700, 12.09121939910151, 12.55605456519257, 13.14574947149285, 13.29933280503938, 13.90943038319876, 14.57692214582206, 14.94220123121196

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