Properties

Label 2-51600-1.1-c1-0-28
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 − 3·7-s + 9-s + 5·11-s + 3·13-s − 7·19-s − 3·21-s − 4·23-s + 27-s + 29-s + 6·31-s + 5·33-s + 6·37-s + 3·39-s + 43-s − 3·47-s + 2·49-s − 12·53-s − 7·57-s + 4·59-s + 12·61-s − 3·63-s + 10·67-s − 4·69-s − 8·71-s + 16·73-s − 15·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.832·13-s − 1.60·19-s − 0.654·21-s − 0.834·23-s + 0.192·27-s + 0.185·29-s + 1.07·31-s + 0.870·33-s + 0.986·37-s + 0.480·39-s + 0.152·43-s − 0.437·47-s + 2/7·49-s − 1.64·53-s − 0.927·57-s + 0.520·59-s + 1.53·61-s − 0.377·63-s + 1.22·67-s − 0.481·69-s − 0.949·71-s + 1.87·73-s − 1.70·77-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

\(L(1)\) \(\approx\) \(2.715598916\)
\(L(\frac12)\) \(\approx\) \(2.715598916\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
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 \( 1 - T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.52791736429971, −13.97717402188115, −13.46956781763163, −12.98625280384114, −12.51997232230350, −12.06532236567636, −11.34561778281204, −10.95265105066996, −10.15914349734909, −9.775992395684325, −9.283100089247861, −8.815963310928188, −8.181320039228330, −7.892152300098166, −6.678637375991654, −6.568524951774452, −6.297316282164912, −5.422567953838954, −4.431497373867461, −3.991593392133347, −3.612072823474743, −2.848238273872081, −2.175268858381489, −1.407628015470235, −0.5704524564065271, 0.5704524564065271, 1.407628015470235, 2.175268858381489, 2.848238273872081, 3.612072823474743, 3.991593392133347, 4.431497373867461, 5.422567953838954, 6.297316282164912, 6.568524951774452, 6.678637375991654, 7.892152300098166, 8.181320039228330, 8.815963310928188, 9.283100089247861, 9.775992395684325, 10.15914349734909, 10.95265105066996, 11.34561778281204, 12.06532236567636, 12.51997232230350, 12.98625280384114, 13.46956781763163, 13.97717402188115, 14.52791736429971

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