Properties

Label 2-189-3.2-c2-0-14
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  − 2.71i·2-s − 3.35·4-s − 7.17i·5-s + 2.64·7-s − 1.75i·8-s − 19.4·10-s + 2.71i·11-s + 6.35·13-s − 7.17i·14-s − 18.1·16-s + 6.38i·17-s − 30.5·19-s + 24.0i·20-s + 7.35·22-s − 27.9i·23-s + ⋯
L(s)  = 1  − 1.35i·2-s − 0.838·4-s − 1.43i·5-s + 0.377·7-s − 0.218i·8-s − 1.94·10-s + 0.246i·11-s + 0.488·13-s − 0.512i·14-s − 1.13·16-s + 0.375i·17-s − 1.60·19-s + 1.20i·20-s + 0.334·22-s − 1.21i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.43999i\)
\(L(\frac12)\) \(\approx\) \(1.43999i\)
\(L(2)\) 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 \( 1 + 2.71iT - 4T^{2} \)
5 \( 1 + 7.17iT - 25T^{2} \)
11 \( 1 - 2.71iT - 121T^{2} \)
13 \( 1 - 6.35T + 169T^{2} \)
17 \( 1 - 6.38iT - 289T^{2} \)
19 \( 1 + 30.5T + 361T^{2} \)
23 \( 1 + 27.9iT - 529T^{2} \)
29 \( 1 - 37.0iT - 841T^{2} \)
31 \( 1 - 44.3T + 961T^{2} \)
37 \( 1 - 71.2T + 1.36e3T^{2} \)
41 \( 1 + 28.8iT - 1.68e3T^{2} \)
43 \( 1 + 4.68T + 1.84e3T^{2} \)
47 \( 1 + 65.7iT - 2.20e3T^{2} \)
53 \( 1 - 36.0iT - 2.80e3T^{2} \)
59 \( 1 + 101. iT - 3.48e3T^{2} \)
61 \( 1 - 45.5T + 3.72e3T^{2} \)
67 \( 1 - 93.8T + 4.48e3T^{2} \)
71 \( 1 - 80.8iT - 5.04e3T^{2} \)
73 \( 1 + 60.2T + 5.32e3T^{2} \)
79 \( 1 - 20.6T + 6.24e3T^{2} \)
83 \( 1 + 94.7iT - 6.88e3T^{2} \)
89 \( 1 - 52.1iT - 7.92e3T^{2} \)
97 \( 1 - 98.2T + 9.40e3T^{2} \)
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

−11.92825338834154841466193264477, −10.92940580717194441243710500061, −10.08605976518209848770528808472, −8.898147393268865166219460007823, −8.310782518013089284413698638405, −6.47630001991422402188450904474, −4.84653302238203726524711875246, −4.02936669833723122438795236836, −2.20247004354222222976786371787, −0.879714582107235026578769306463, 2.60293449140380309263588697661, 4.34240232919681319874182734348, 5.95181494010663835341818846769, 6.52410439288956774032694763610, 7.59914572549747203251965824758, 8.347952608025344022939181259992, 9.745073113974419101645705576085, 10.98640995386590556495198412568, 11.53764317046855470659468613967, 13.31423876686308478108256872919

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