↑

Properties

Label 2-10890-1.1-c1-0-18
Degree $2$
Conductor $10890$
Sign $1$
Analytic cond. $86.9570$
Root an. cond. $9.32507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s − 6·13-s + 16-s + 6·17-s + 20-s + 6·23-s + 25-s − 6·26-s + 4·31-s + 32-s + 6·34-s − 2·37-s + 40-s + 12·41-s + 12·43-s + 6·46-s − 6·47-s − 7·49-s + 50-s − 6·52-s − 6·53-s + 4·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.66·13-s + 1/4·16-s + 1.45·17-s + 0.223·20-s + 1.25·23-s + 1/5·25-s − 1.17·26-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.158·40-s + 1.87·41-s + 1.82·43-s + 0.884·46-s − 0.875·47-s − 49-s + 0.141·50-s − 0.832·52-s − 0.824·53-s + 0.508·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10890\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(86.9570\)
Root analytic conductor: \(9.32507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10890,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.829480207\)
\(L(\frac12)\) \(\approx\) \(3.829480207\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−16.47485359521099, −15.98061106983294, −15.14160161771121, −14.71068467323880, −14.25853975733501, −13.83080587019365, −12.88256251320329, −12.59890196876633, −12.12328646745091, −11.36222206810950, −10.79131546667575, −10.02182744579332, −9.643656356170030, −8.986444007451215, −7.972998004259617, −7.458860056420024, −6.952746523493199, −6.029820533520251, −5.571722664612665, −4.799242686484448, −4.381835932564793, −3.141020330725166, −2.826321435124352, −1.852177541231141, −0.8241041219040363, 0.8241041219040363, 1.852177541231141, 2.826321435124352, 3.141020330725166, 4.381835932564793, 4.799242686484448, 5.571722664612665, 6.029820533520251, 6.952746523493199, 7.458860056420024, 7.972998004259617, 8.986444007451215, 9.643656356170030, 10.02182744579332, 10.79131546667575, 11.36222206810950, 12.12328646745091, 12.59890196876633, 12.88256251320329, 13.83080587019365, 14.25853975733501, 14.71068467323880, 15.14160161771121, 15.98061106983294, 16.47485359521099

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