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Properties

Label 2-189-7.4-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

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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: $\chi_{189} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43149 - 0.182420i\)
\(L(\frac12)\) \(\approx\) \(1.43149 - 0.182420i\)
\(L(\frac{3}{2})\) 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 \( 1 - 9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.66325444120093442589088370226, −11.52135417211737529592403696610, −10.91163410613688514849341903148, −9.097171371402099595716051490550, −8.491108875433920823632139901996, −7.46298497246863215298517608625, −5.98028871689103106929561727479, −5.06267473736550086588148582320, −4.14387406148145574160586639230, −1.48624052642395739323500039129, 2.36587211598253029337752680266, 3.63529057435228304430447279733, 4.52109332697467195722041791113, 6.77741518484778729195316987935, 7.31358002864404554212208615440, 8.300079537147651787454175107691, 10.14691296986573405702794469107, 11.02764885332401663289194776595, 11.40145777942347560492962968641, 12.34746182005289072543014347442

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