↑

Properties

Label 2-189-7.3-c0-0-0
Degree $2$
Conductor $189$
Sign $0.922 + 0.386i$
Analytic cond. $0.0943232$
Root an. cond. $0.307120$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)4-s − 7-s + 1.73i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + 49-s + (1.49 + 0.866i)52-s + (1.5 − 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s − 7-s + 1.73i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + 49-s + (1.49 + 0.866i)52-s + (1.5 − 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(0.0943232\)
Root analytic conductor: \(0.307120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :0),\ 0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6930928372\)
\(L(\frac12)\) \(\approx\) \(0.6930928372\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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

−12.73239221954578803744320670572, −11.61762879820832239756966354126, −10.85521901287101135048632228002, −9.626068553937824179479037158025, −9.165654541498292184752169336727, −7.31006683125025324870153340290, −6.51542422563339517944929414863, −5.48776915957869355677584670108, −3.88563800900704074452408537378, −2.08500484518962646615119309951, 2.74706031030906967687092263124, 3.75222531661024383459138765557, 5.59099571872574271090089067630, 6.75727971220309298784070821063, 7.75267689373509500079668521396, 8.729187256536063775811934105183, 10.03256821860987001826954254153, 10.87364776718320672025700466999, 12.14654922780244633590250917269, 12.74477142285516605760857436266

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