↑

Properties

Label 2-96600-1.1-c1-0-31
Degree $2$
Conductor $96600$
Sign $-1$
Analytic cond. $771.354$
Root an. cond. $27.7732$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 4·11-s − 2·13-s + 2·17-s − 4·19-s + 21-s + 23-s − 27-s + 2·29-s + 4·33-s − 6·37-s + 2·39-s − 6·41-s + 8·43-s − 4·47-s + 49-s − 2·51-s − 2·53-s + 4·57-s + 12·59-s − 2·61-s − 63-s − 8·67-s − 69-s + 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.125·63-s − 0.977·67-s − 0.120·69-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(771.354\)
Root analytic conductor: \(27.7732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.96005276207442, −13.46909683034004, −12.93170219707743, −12.58914498119439, −12.16205971752074, −11.63236485690073, −10.97853217762690, −10.59121909686496, −10.14696367748052, −9.769412447277300, −9.100002285264648, −8.489104244659400, −8.008706579152522, −7.473894426975145, −6.865720885195545, −6.525828770615348, −5.727239265593064, −5.392298199616375, −4.843107742795018, −4.281142845206209, −3.567497349699231, −2.912151207375400, −2.342012001773986, −1.647191279693508, −0.6664250050227764, 0, 0.6664250050227764, 1.647191279693508, 2.342012001773986, 2.912151207375400, 3.567497349699231, 4.281142845206209, 4.843107742795018, 5.392298199616375, 5.727239265593064, 6.525828770615348, 6.865720885195545, 7.473894426975145, 8.008706579152522, 8.489104244659400, 9.100002285264648, 9.769412447277300, 10.14696367748052, 10.59121909686496, 10.97853217762690, 11.63236485690073, 12.16205971752074, 12.58914498119439, 12.93170219707743, 13.46909683034004, 13.96005276207442

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