Properties

Label 2-15600-1.1-c1-0-34
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 − 2·11-s − 13-s + 2·17-s + 4·19-s + 2·21-s − 27-s + 4·29-s − 8·31-s + 2·33-s + 6·37-s + 39-s − 6·41-s − 4·43-s + 8·47-s − 3·49-s − 2·51-s + 2·53-s − 4·57-s − 10·59-s − 14·61-s − 2·63-s + 16·67-s + 4·71-s + 8·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 0.436·21-s − 0.192·27-s + 0.742·29-s − 1.43·31-s + 0.348·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.280·51-s + 0.274·53-s − 0.529·57-s − 1.30·59-s − 1.79·61-s − 0.251·63-s + 1.95·67-s + 0.474·71-s + 0.936·73-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−16.29305919863156, −15.75691742044696, −15.36448701180490, −14.64402674393646, −13.95670679764539, −13.49978868880629, −12.76513699878150, −12.46142067342781, −11.83564431051341, −11.23359760207958, −10.57926997457107, −10.11705507126786, −9.479411888718195, −9.074250272151562, −8.009550670714733, −7.659565545807637, −6.873803411987376, −6.383413411542359, −5.573635505151433, −5.196032410453200, −4.406033872592156, −3.500034670468278, −2.977337079578809, −2.015777699180566, −0.9642452441243227, 0, 0.9642452441243227, 2.015777699180566, 2.977337079578809, 3.500034670468278, 4.406033872592156, 5.196032410453200, 5.573635505151433, 6.383413411542359, 6.873803411987376, 7.659565545807637, 8.009550670714733, 9.074250272151562, 9.479411888718195, 10.11705507126786, 10.57926997457107, 11.23359760207958, 11.83564431051341, 12.46142067342781, 12.76513699878150, 13.49978868880629, 13.95670679764539, 14.64402674393646, 15.36448701180490, 15.75691742044696, 16.29305919863156

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