Properties

Label 2-189-7.2-c1-0-6
Degree $2$
Conductor $189$
Sign $0.968 - 0.250i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
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)2-s + (0.500 + 0.866i)4-s + (2 − 3.46i)5-s + (2 − 1.73i)7-s − 3·8-s + (1.99 + 3.46i)10-s + (−1 − 1.73i)11-s + 13-s + (0.499 + 2.59i)14-s + (0.500 − 0.866i)16-s + (3 + 5.19i)17-s + (−2 + 3.46i)19-s + 4·20-s + 1.99·22-s + (−3 + 5.19i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s + (0.894 − 1.54i)5-s + (0.755 − 0.654i)7-s − 1.06·8-s + (0.632 + 1.09i)10-s + (−0.301 − 0.522i)11-s + 0.277·13-s + (0.133 + 0.694i)14-s + (0.125 − 0.216i)16-s + (0.727 + 1.26i)17-s + (−0.458 + 0.794i)19-s + 0.894·20-s + 0.426·22-s + (−0.625 + 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20876 + 0.154037i\)
\(L(\frac12)\) \(\approx\) \(1.20876 + 0.154037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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

−12.68522400706573042808288873370, −11.81025057505638265883708592369, −10.48024661562410809117275995165, −9.373838237260404718452307206033, −8.246535930921999168223185001496, −7.966233014049066905849841644330, −6.22333767532063700394517592893, −5.38175250394855804596159596545, −3.89375069261998848412767326432, −1.58396956880540069348832040896, 2.06956961744991906432622329790, 2.87627368937269559306127638353, 5.20866446729070686428160561998, 6.27570760253699139758996857516, 7.22406986068622315384115031300, 8.820695427120424241574013092030, 9.902330088778423068787203028341, 10.51661116892570110161586728721, 11.32705007789106230782260618041, 12.15426450409599511608426842844

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