Properties

Label 2-51600-1.1-c1-0-15
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 − 11-s − 5·13-s + 3·17-s − 4·21-s + 3·23-s − 27-s − 8·29-s + 31-s + 33-s − 8·37-s + 5·39-s − 41-s − 43-s + 9·49-s − 3·51-s − 3·53-s + 8·61-s + 4·63-s − 9·67-s − 3·69-s − 4·71-s − 4·73-s − 4·77-s − 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.727·17-s − 0.872·21-s + 0.625·23-s − 0.192·27-s − 1.48·29-s + 0.179·31-s + 0.174·33-s − 1.31·37-s + 0.800·39-s − 0.156·41-s − 0.152·43-s + 9/7·49-s − 0.420·51-s − 0.412·53-s + 1.02·61-s + 0.503·63-s − 1.09·67-s − 0.361·69-s − 0.474·71-s − 0.468·73-s − 0.455·77-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

\(L(1)\) \(\approx\) \(1.609792604\)
\(L(\frac12)\) \(\approx\) \(1.609792604\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−14.40864891023650, −14.25953157521378, −13.37557131950819, −12.92341261415964, −12.32967275553776, −11.78616036518228, −11.55705784998953, −10.90233777397990, −10.41373395764156, −9.989232772012115, −9.282056085456019, −8.752488396173857, −8.060760036873995, −7.524814728680606, −7.288914887499493, −6.564221497561844, −5.643004343585459, −5.315220361455454, −4.871561143574158, −4.341191801037110, −3.537314217433900, −2.737793951256937, −1.929191330552354, −1.468745068084233, −0.4595361651373856, 0.4595361651373856, 1.468745068084233, 1.929191330552354, 2.737793951256937, 3.537314217433900, 4.341191801037110, 4.871561143574158, 5.315220361455454, 5.643004343585459, 6.564221497561844, 7.288914887499493, 7.524814728680606, 8.060760036873995, 8.752488396173857, 9.282056085456019, 9.989232772012115, 10.41373395764156, 10.90233777397990, 11.55705784998953, 11.78616036518228, 12.32967275553776, 12.92341261415964, 13.37557131950819, 14.25953157521378, 14.40864891023650

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