Properties

Label 2-51600-1.1-c1-0-51
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 + 9-s − 4·11-s + 6·13-s + 6·17-s + 8·19-s − 4·23-s + 27-s + 10·29-s − 4·33-s + 6·37-s + 6·39-s − 6·41-s + 43-s + 12·47-s − 7·49-s + 6·51-s + 6·53-s + 8·57-s + 4·59-s − 6·61-s + 4·67-s − 4·69-s − 8·71-s + 10·73-s + 8·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 1.45·17-s + 1.83·19-s − 0.834·23-s + 0.192·27-s + 1.85·29-s − 0.696·33-s + 0.986·37-s + 0.960·39-s − 0.937·41-s + 0.152·43-s + 1.75·47-s − 49-s + 0.840·51-s + 0.824·53-s + 1.05·57-s + 0.520·59-s − 0.768·61-s + 0.488·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-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

\(L(1)\) \(\approx\) \(4.256653050\)
\(L(\frac12)\) \(\approx\) \(4.256653050\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.25334165810763, −13.86053434048880, −13.71645949215626, −13.04243393650398, −12.47177228456956, −11.94469874193214, −11.48516971341421, −10.76914935398735, −10.23874915496893, −9.922688528858054, −9.334006155698718, −8.544467102473249, −8.238396519809237, −7.723388587504086, −7.292912735840633, −6.475751473359281, −5.810267679132789, −5.458194675550600, −4.741884573171524, −3.995031867895881, −3.279625946356175, −3.031561825766237, −2.206438564032547, −1.219465026012271, −0.8085781758914221, 0.8085781758914221, 1.219465026012271, 2.206438564032547, 3.031561825766237, 3.279625946356175, 3.995031867895881, 4.741884573171524, 5.458194675550600, 5.810267679132789, 6.475751473359281, 7.292912735840633, 7.723388587504086, 8.238396519809237, 8.544467102473249, 9.334006155698718, 9.922688528858054, 10.23874915496893, 10.76914935398735, 11.48516971341421, 11.94469874193214, 12.47177228456956, 13.04243393650398, 13.71645949215626, 13.86053434048880, 14.25334165810763

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