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Properties

Label 2-189-63.59-c1-0-1
Degree $2$
Conductor $189$
Sign $-0.102 + 0.994i$
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  + (−1.30 − 0.755i)2-s + (0.140 + 0.242i)4-s + 0.775·5-s + (2.05 − 1.66i)7-s + 2.59i·8-s + (−1.01 − 0.585i)10-s − 3.84i·11-s + (−2.54 − 1.46i)13-s + (−3.94 + 0.618i)14-s + (2.24 − 3.88i)16-s + (2.69 − 4.67i)17-s + (−0.376 + 0.217i)19-s + (0.108 + 0.188i)20-s + (−2.90 + 5.02i)22-s + 0.0557i·23-s + ⋯
L(s)  = 1  + (−0.924 − 0.533i)2-s + (0.0700 + 0.121i)4-s + 0.346·5-s + (0.778 − 0.628i)7-s + 0.918i·8-s + (−0.320 − 0.185i)10-s − 1.15i·11-s + (−0.705 − 0.407i)13-s + (−1.05 + 0.165i)14-s + (0.560 − 0.970i)16-s + (0.654 − 1.13i)17-s + (−0.0863 + 0.0498i)19-s + (0.0243 + 0.0421i)20-s + (−0.618 + 1.07i)22-s + 0.0116i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.102 + 0.994i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.102 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506619 - 0.561749i\)
\(L(\frac12)\) \(\approx\) \(0.506619 - 0.561749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 + 1.66i)T \)
good2 \( 1 + (1.30 + 0.755i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 0.775T + 5T^{2} \)
11 \( 1 + 3.84iT - 11T^{2} \)
13 \( 1 + (2.54 + 1.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.69 + 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.376 - 0.217i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.0557iT - 23T^{2} \)
29 \( 1 + (-0.187 + 0.108i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.67 + 3.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.14 - 5.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.78 - 6.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.42 - 11.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.482 - 0.836i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.46 - 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.56 - 2.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.01 + 1.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.10 - 3.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.50iT - 71T^{2} \)
73 \( 1 + (7.05 + 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.48 + 4.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.31 - 7.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.82 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.24 + 0.716i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.80855761463712147621356529204, −11.23955012391632456627275784002, −10.19041314190746080816953269966, −9.560865380021121211747960018954, −8.316322281229163248627094878838, −7.62136909348754346428887873460, −5.89367603664415413772981148397, −4.74891142152707397477515615259, −2.76903051701591595685329234915, −1.00265109611426784812380914614, 1.97917685175824363867946545380, 4.17653634802018374766490440875, 5.58399543994196542256751562337, 6.93711554271953703653340989381, 7.84361770321641859637549461277, 8.758318931216808598432432221485, 9.689657705291254644847103847945, 10.47739493587574512400707478673, 12.03766750481526750519760655974, 12.56051580069189089924334482491

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