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Properties

Label 2-189-63.5-c1-0-1
Degree $2$
Conductor $189$
Sign $-0.749 - 0.662i$
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  + 2.09i·2-s − 2.39·4-s + (1.04 + 1.80i)5-s + (2.60 + 0.486i)7-s − 0.819i·8-s + (−3.79 + 2.18i)10-s + (−2.79 − 1.61i)11-s + (−2.68 − 1.55i)13-s + (−1.01 + 5.44i)14-s − 3.06·16-s + (−0.816 − 1.41i)17-s + (4.79 + 2.76i)19-s + (−2.49 − 4.32i)20-s + (3.38 − 5.85i)22-s + (1.00 − 0.580i)23-s + ⋯
L(s)  = 1  + 1.48i·2-s − 1.19·4-s + (0.467 + 0.809i)5-s + (0.982 + 0.183i)7-s − 0.289i·8-s + (−1.19 + 0.692i)10-s + (−0.843 − 0.486i)11-s + (−0.745 − 0.430i)13-s + (−0.272 + 1.45i)14-s − 0.766·16-s + (−0.197 − 0.342i)17-s + (1.09 + 0.634i)19-s + (−0.558 − 0.967i)20-s + (0.721 − 1.24i)22-s + (0.209 − 0.121i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.749 - 0.662i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445835 + 1.17786i\)
\(L(\frac12)\) \(\approx\) \(0.445835 + 1.17786i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.486i)T \)
good2 \( 1 - 2.09iT - 2T^{2} \)
5 \( 1 + (-1.04 - 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.79 + 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.68 + 1.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.816 + 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.79 - 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.00 + 0.580i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.05 + 4.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.96iT - 31T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.35 - 2.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.974 + 1.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.13T + 47T^{2} \)
53 \( 1 + (-5.27 + 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.96T + 59T^{2} \)
61 \( 1 + 4.79iT - 61T^{2} \)
67 \( 1 + 0.673T + 67T^{2} \)
71 \( 1 + 7.01iT - 71T^{2} \)
73 \( 1 + (2.96 - 1.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + (-1.54 - 2.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.45 - 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.07 - 1.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.45107287835890323355474756383, −11.96588635213837141393666992112, −10.86615629189624096585124751841, −9.868237459701506397128339184824, −8.458006948627170622249291502005, −7.74887611743675391720947098930, −6.79934970092097712344464255571, −5.64700415049810561167061765936, −4.88592622808414784242912477523, −2.67116898504546692365944055809, 1.36204481387660754923117448507, 2.65687486688240048697128031327, 4.47907152117764938275520614692, 5.18010831137235352432774634915, 7.19234877425529491462492095542, 8.503050981494284095162200129509, 9.536922917017560998927131652154, 10.28417760775648183114612332872, 11.34274812008040481118658796332, 12.03451114114315860776524155473

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