↑

Properties

Label 1-189-189.124-r1-0-0
Degree $1$
Conductor $189$
Sign $0.871 + 0.490i$
Analytic cond. $20.3108$
Root an. cond. $20.3108$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (0.173 + 0.984i)11-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (0.173 + 0.984i)11-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ 0.871 + 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.402165638 + 0.6302950235i\)
\(L(\frac12)\) \(\approx\) \(2.402165638 + 0.6302950235i\)
\(L(1)\) \(\approx\) \(1.553419084 - 0.09551605035i\)
\(L(1)\) \(\approx\) \(1.553419084 - 0.09551605035i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.90563242253770922205069576242, −25.46490692706634829582714858679, −24.7342671602552899427910086954, −24.15394642437255500484157452951, −23.00473366285007488640486100101, −22.30652862469270197539055233548, −20.94436808266979748120146402421, −20.57735319030597111449905819747, −19.183312805785704283253303712105, −17.82213900680127703009942442768, −16.7058524094133228820389951673, −16.20392022839895841695189896542, −15.11532663820722531690851936800, −13.99778216654786326197604504575, −13.09339612600449933143942529226, −12.224461390406194015425356893652, −11.24605993863070321912313503637, −9.52336265228318589640379510970, −8.31571779932271780089807136664, −7.604574972818520960266311863976, −6.006101774667666809956357017335, −5.24903893218116158775153670338, −4.05404867899855305448320113050, −2.839176358197596474598741170754, −0.70007801809846050014950728935, 1.55200144116412771359624861186, 2.808403371923576365443196960955, 3.91936336732783163961262675983, 5.087920189368766899301603314819, 6.52253213358446106963045780250, 7.28197650202416113878743444472, 9.22440358607148962366055732665, 10.21852673370729778909178223308, 11.19571841593941326221406767389, 12.043061761180392917459942239186, 13.141689942626422871378032351037, 14.33468981380280657416378872127, 14.87448192788533261761126046962, 15.9218844840820447371693858697, 17.49006955903672343245173099987, 18.568281026440065913743292210524, 19.4198370385215621965990757400, 20.21797237727861782496020375544, 21.60328374700864799413552190330, 21.95844780452442314749669254529, 23.27018894785498749264096221667, 23.58726872415782785679606089665, 25.01813182513319350302139750493, 25.99809375759171075791115679198, 27.07992353271348224394050063611

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