Properties

Label 2-15600-1.1-c1-0-48
Degree $2$
Conductor $15600$
Sign $-1$
Analytic cond. $124.566$
Root an. cond. $11.1609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·7-s + 9-s − 3·11-s − 13-s − 3·17-s − 3·21-s + 3·23-s − 27-s + 8·29-s − 4·31-s + 3·33-s − 37-s + 39-s − 3·41-s − 4·43-s − 10·47-s + 2·49-s + 3·51-s + 9·53-s − 4·59-s + 9·61-s + 3·63-s + 4·67-s − 3·69-s − 7·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.727·17-s − 0.654·21-s + 0.625·23-s − 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.522·33-s − 0.164·37-s + 0.160·39-s − 0.468·41-s − 0.609·43-s − 1.45·47-s + 2/7·49-s + 0.420·51-s + 1.23·53-s − 0.520·59-s + 1.15·61-s + 0.377·63-s + 0.488·67-s − 0.361·69-s − 0.830·71-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(124.566\)
Root analytic conductor: \(11.1609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 15600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
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

−16.24661725823891, −15.80094035897652, −15.07365327527106, −14.78411427458426, −14.07139196375910, −13.36952070961480, −13.03998019603514, −12.24058884816358, −11.75018810275777, −11.22934032196111, −10.64669540540418, −10.28994595963119, −9.494194336272716, −8.747935040678532, −8.158289171580181, −7.712616216061724, −6.857043788749007, −6.480493549032704, −5.407524097492107, −5.064137884764465, −4.593258399334252, −3.698988262564904, −2.712686293253548, −1.996656211672393, −1.110721879550951, 0, 1.110721879550951, 1.996656211672393, 2.712686293253548, 3.698988262564904, 4.593258399334252, 5.064137884764465, 5.407524097492107, 6.480493549032704, 6.857043788749007, 7.712616216061724, 8.158289171580181, 8.747935040678532, 9.494194336272716, 10.28994595963119, 10.64669540540418, 11.22934032196111, 11.75018810275777, 12.24058884816358, 13.03998019603514, 13.36952070961480, 14.07139196375910, 14.78411427458426, 15.07365327527106, 15.80094035897652, 16.24661725823891

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