Properties

Label 2-18240-1.1-c1-0-54
Degree $2$
Conductor $18240$
Sign $-1$
Analytic cond. $145.647$
Root an. cond. $12.0684$
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 + 5-s + 9-s + 4·11-s − 2·13-s − 15-s + 2·17-s − 19-s − 4·23-s + 25-s − 27-s − 6·29-s − 4·31-s − 4·33-s + 6·37-s + 2·39-s + 10·41-s − 4·43-s + 45-s + 12·47-s − 7·49-s − 2·51-s − 6·53-s + 4·55-s + 57-s − 12·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s + 0.986·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.75·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.539·55-s + 0.132·57-s − 1.56·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18240\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(145.647\)
Root analytic conductor: \(12.0684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{18240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 18240,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−16.21630317879920, −15.56733433112491, −14.78238738038503, −14.48376114613374, −13.98583976464254, −13.22307764353756, −12.65980409831002, −12.23359876233312, −11.60954711490558, −11.12451737019382, −10.52104055371817, −9.837798398643502, −9.347754813642484, −8.978584983184948, −7.936449176787951, −7.484242359555010, −6.800527079617705, −6.035277241618493, −5.838949810580975, −4.955757571760503, −4.245081412824579, −3.714945252623037, −2.718764893085116, −1.850913936294586, −1.161491410029140, 0, 1.161491410029140, 1.850913936294586, 2.718764893085116, 3.714945252623037, 4.245081412824579, 4.955757571760503, 5.838949810580975, 6.035277241618493, 6.800527079617705, 7.484242359555010, 7.936449176787951, 8.978584983184948, 9.347754813642484, 9.837798398643502, 10.52104055371817, 11.12451737019382, 11.60954711490558, 12.23359876233312, 12.65980409831002, 13.22307764353756, 13.98583976464254, 14.48376114613374, 14.78238738038503, 15.56733433112491, 16.21630317879920

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