Properties

Label 2-51600-1.1-c1-0-16
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 + 4·13-s − 2·17-s − 2·19-s + 3·21-s − 7·23-s − 27-s + 6·29-s + 6·31-s + 10·37-s − 4·39-s − 2·41-s + 43-s + 7·47-s + 2·49-s + 2·51-s + 14·53-s + 2·57-s − 8·59-s − 2·61-s − 3·63-s − 11·67-s + 7·69-s − 4·71-s + 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.10·13-s − 0.485·17-s − 0.458·19-s + 0.654·21-s − 1.45·23-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 1.64·37-s − 0.640·39-s − 0.312·41-s + 0.152·43-s + 1.02·47-s + 2/7·49-s + 0.280·51-s + 1.92·53-s + 0.264·57-s − 1.04·59-s − 0.256·61-s − 0.377·63-s − 1.34·67-s + 0.842·69-s − 0.474·71-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

\(L(1)\) \(\approx\) \(1.357503687\)
\(L(\frac12)\) \(\approx\) \(1.357503687\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−14.45239249572482, −13.71426301444087, −13.48756128012920, −13.02264328160585, −12.35015990541163, −11.96020492209113, −11.52168430905685, −10.75402923610970, −10.35719692640160, −10.01727797053433, −9.241226938046537, −8.877766657940908, −8.138013689912603, −7.704267104938600, −6.816111599551602, −6.393491438228034, −6.070723036137018, −5.578307761660646, −4.529111552373506, −4.238283719808830, −3.551852813498680, −2.813922923243842, −2.181541643523047, −1.169821457692164, −0.4687393084426406, 0.4687393084426406, 1.169821457692164, 2.181541643523047, 2.813922923243842, 3.551852813498680, 4.238283719808830, 4.529111552373506, 5.578307761660646, 6.070723036137018, 6.393491438228034, 6.816111599551602, 7.704267104938600, 8.138013689912603, 8.877766657940908, 9.241226938046537, 10.01727797053433, 10.35719692640160, 10.75402923610970, 11.52168430905685, 11.96020492209113, 12.35015990541163, 13.02264328160585, 13.48756128012920, 13.71426301444087, 14.45239249572482

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