↑

Properties

Label 2-189-3.2-c2-0-10
Degree $2$
Conductor $189$
Sign $1$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.55i·2-s − 8.64·4-s − 9.40i·5-s − 2.64·7-s − 16.5i·8-s + 33.4·10-s − 3.55i·11-s + 11.6·13-s − 9.40i·14-s + 24.1·16-s − 27.1i·17-s + 6.52·19-s + 81.3i·20-s + 12.6·22-s − 1.03i·23-s + ⋯
L(s)  = 1  + 1.77i·2-s − 2.16·4-s − 1.88i·5-s − 0.377·7-s − 2.06i·8-s + 3.34·10-s − 0.323i·11-s + 0.895·13-s − 0.672i·14-s + 1.51·16-s − 1.59i·17-s + 0.343·19-s + 4.06i·20-s + 0.574·22-s − 0.0450i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03031\)
\(L(\frac12)\) \(\approx\) \(1.03031\)
\(L(2)\) 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 \( 1 + 2.64T \)
good2 \( 1 - 3.55iT - 4T^{2} \)
5 \( 1 + 9.40iT - 25T^{2} \)
11 \( 1 + 3.55iT - 121T^{2} \)
13 \( 1 - 11.6T + 169T^{2} \)
17 \( 1 + 27.1iT - 289T^{2} \)
19 \( 1 - 6.52T + 361T^{2} \)
23 \( 1 + 1.03iT - 529T^{2} \)
29 \( 1 + 29.7iT - 841T^{2} \)
31 \( 1 + 24.3T + 961T^{2} \)
37 \( 1 + 45.2T + 1.36e3T^{2} \)
41 \( 1 - 19.0iT - 1.68e3T^{2} \)
43 \( 1 - 74.6T + 1.84e3T^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + 9.63iT - 2.80e3T^{2} \)
59 \( 1 - 1.70iT - 3.48e3T^{2} \)
61 \( 1 - 98.4T + 3.72e3T^{2} \)
67 \( 1 + 27.8T + 4.48e3T^{2} \)
71 \( 1 - 63.3iT - 5.04e3T^{2} \)
73 \( 1 + 33.7T + 5.32e3T^{2} \)
79 \( 1 + 58.6T + 6.24e3T^{2} \)
83 \( 1 - 67.7iT - 6.88e3T^{2} \)
89 \( 1 - 25.7iT - 7.92e3T^{2} \)
97 \( 1 + 18.2T + 9.40e3T^{2} \)
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

−12.80728321733400213050738987263, −11.63120039818727981361429120856, −9.637122761009663942800379056683, −8.978670741285576359503607000797, −8.267307339652795593814267019001, −7.22430120390574426413990746844, −5.87100154760007751006762718250, −5.17795544687596938370658913589, −4.09686092634987802290807770820, −0.62882790953537701146558604373, 1.90906493121477313643922568113, 3.21017752843723406380439389958, 3.88309497078710048349157427112, 5.92803095851490694075688131557, 7.18851807216512888061728722536, 8.712922005625678938926573025888, 9.901836528619403342481387054949, 10.69596835691072811900939199908, 11.00039910549897188283289166795, 12.15781763552038641818258413167

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