Properties

Label 2-15600-1.1-c1-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 5·11-s − 13-s − 3·17-s + 6·19-s − 21-s + 7·23-s + 27-s − 6·29-s + 10·31-s + 5·33-s + 11·37-s − 39-s + 5·41-s − 4·43-s − 2·47-s − 6·49-s − 3·51-s + 5·53-s + 6·57-s + 61-s − 63-s − 4·67-s + 7·69-s + 5·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s − 0.727·17-s + 1.37·19-s − 0.218·21-s + 1.45·23-s + 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.870·33-s + 1.80·37-s − 0.160·39-s + 0.780·41-s − 0.609·43-s − 0.291·47-s − 6/7·49-s − 0.420·51-s + 0.686·53-s + 0.794·57-s + 0.128·61-s − 0.125·63-s − 0.488·67-s + 0.842·69-s + 0.593·71-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: \(0\)
Selberg data: \((2,\ 15600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.303400950\)
\(L(\frac12)\) \(\approx\) \(3.303400950\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.93321568462797, −15.38035198792503, −14.82097794499665, −14.40543736246598, −13.83020814724607, −13.12604425966132, −12.93173471697951, −11.95846593851806, −11.53406908488669, −11.12142010194859, −10.08996605364885, −9.660378179118891, −9.135650141174280, −8.758627507559483, −7.855438611410937, −7.329293235862716, −6.653456434908869, −6.224934637536782, −5.293355154517723, −4.515741773115546, −3.969266371570333, −3.128499222946846, −2.645251617914425, −1.524429728192594, −0.8243522546221290, 0.8243522546221290, 1.524429728192594, 2.645251617914425, 3.128499222946846, 3.969266371570333, 4.515741773115546, 5.293355154517723, 6.224934637536782, 6.653456434908869, 7.329293235862716, 7.855438611410937, 8.758627507559483, 9.135650141174280, 9.660378179118891, 10.08996605364885, 11.12142010194859, 11.53406908488669, 11.95846593851806, 12.93173471697951, 13.12604425966132, 13.83020814724607, 14.40543736246598, 14.82097794499665, 15.38035198792503, 15.93321568462797

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