Properties

Label 2-15600-1.1-c1-0-51
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 + 4·7-s + 9-s − 2·11-s − 13-s − 4·17-s − 2·19-s − 4·21-s − 6·23-s − 27-s − 2·29-s + 4·31-s + 2·33-s + 6·37-s + 39-s − 6·41-s + 8·43-s + 8·47-s + 9·49-s + 4·51-s − 10·53-s + 2·57-s + 14·59-s + 10·61-s + 4·63-s + 4·67-s + 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.970·17-s − 0.458·19-s − 0.872·21-s − 1.25·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.16·47-s + 9/7·49-s + 0.560·51-s − 1.37·53-s + 0.264·57-s + 1.82·59-s + 1.28·61-s + 0.503·63-s + 0.488·67-s + 0.722·69-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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−16.26909986004062, −15.68360115196237, −15.23194991804380, −14.61628375453538, −14.11593144719150, −13.51284205494794, −12.89488699158100, −12.30962864353810, −11.61022564016755, −11.32652100690034, −10.70657931994135, −10.22047469803204, −9.534702478730172, −8.686133455133390, −8.187407710181091, −7.684251668392262, −7.020082634558901, −6.256574386556518, −5.607961984857472, −5.012483181340725, −4.392405480703395, −3.932974821036914, −2.532427670405653, −2.086395457845236, −1.122992970630529, 0, 1.122992970630529, 2.086395457845236, 2.532427670405653, 3.932974821036914, 4.392405480703395, 5.012483181340725, 5.607961984857472, 6.256574386556518, 7.020082634558901, 7.684251668392262, 8.187407710181091, 8.686133455133390, 9.534702478730172, 10.22047469803204, 10.70657931994135, 11.32652100690034, 11.61022564016755, 12.30962864353810, 12.89488699158100, 13.51284205494794, 14.11593144719150, 14.61628375453538, 15.23194991804380, 15.68360115196237, 16.26909986004062

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