Properties

Label 2-15600-1.1-c1-0-33
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 − 2·7-s + 9-s − 4·11-s + 13-s − 4·17-s + 2·19-s + 2·21-s + 2·23-s − 27-s + 8·29-s − 4·31-s + 4·33-s − 6·37-s − 39-s + 10·41-s + 4·43-s − 3·49-s + 4·51-s − 6·53-s − 2·57-s + 12·59-s − 2·61-s − 2·63-s − 8·67-s − 2·69-s + 8·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.970·17-s + 0.458·19-s + 0.436·21-s + 0.417·23-s − 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.696·33-s − 0.986·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s − 3/7·49-s + 0.560·51-s − 0.824·53-s − 0.264·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.977·67-s − 0.240·69-s + 0.911·77-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−16.11063012419892, −15.75766159922528, −15.54235374723704, −14.60123508845819, −13.97207485393062, −13.39539537096922, −12.75560011265924, −12.64032266155548, −11.74587548478795, −11.14569306251450, −10.66802336738117, −10.14177140970286, −9.545976620415548, −8.849982147218218, −8.291309816241904, −7.410876665751790, −7.034132949850006, −6.218047306963367, −5.804784231358029, −4.975285042179312, −4.510048795816973, −3.542793994011505, −2.861026258374294, −2.110387980136253, −0.9152025576311207, 0, 0.9152025576311207, 2.110387980136253, 2.861026258374294, 3.542793994011505, 4.510048795816973, 4.975285042179312, 5.804784231358029, 6.218047306963367, 7.034132949850006, 7.410876665751790, 8.291309816241904, 8.849982147218218, 9.545976620415548, 10.14177140970286, 10.66802336738117, 11.14569306251450, 11.74587548478795, 12.64032266155548, 12.75560011265924, 13.39539537096922, 13.97207485393062, 14.60123508845819, 15.54235374723704, 15.75766159922528, 16.11063012419892

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