Properties

Label 2-189-27.16-c1-0-0
Degree $2$
Conductor $189$
Sign $-0.414 - 0.910i$
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  + (−1.88 − 1.58i)2-s + (0.279 + 1.70i)3-s + (0.705 + 4.00i)4-s + (−1.28 − 0.468i)5-s + (2.17 − 3.66i)6-s + (0.173 − 0.984i)7-s + (2.54 − 4.40i)8-s + (−2.84 + 0.954i)9-s + (1.68 + 2.92i)10-s + (−3.59 + 1.30i)11-s + (−6.64 + 2.32i)12-s + (−3.54 + 2.97i)13-s + (−1.88 + 1.58i)14-s + (0.441 − 2.33i)15-s + (−4.13 + 1.50i)16-s + (−1.43 − 2.48i)17-s + ⋯
L(s)  = 1  + (−1.33 − 1.11i)2-s + (0.161 + 0.986i)3-s + (0.352 + 2.00i)4-s + (−0.575 − 0.209i)5-s + (0.889 − 1.49i)6-s + (0.0656 − 0.372i)7-s + (0.899 − 1.55i)8-s + (−0.948 + 0.318i)9-s + (0.533 + 0.923i)10-s + (−1.08 + 0.394i)11-s + (−1.91 + 0.670i)12-s + (−0.982 + 0.824i)13-s + (−0.504 + 0.423i)14-s + (0.114 − 0.601i)15-s + (−1.03 + 0.376i)16-s + (−0.347 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.414 - 0.910i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.414 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111308 + 0.172981i\)
\(L(\frac12)\) \(\approx\) \(0.111308 + 0.172981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.279 - 1.70i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
good2 \( 1 + (1.88 + 1.58i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (1.28 + 0.468i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (3.59 - 1.30i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (3.54 - 2.97i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.43 + 2.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.13 - 3.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.00 - 5.69i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.29 - 4.44i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.0783 + 0.444i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.07 + 5.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.69 - 7.29i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.96 + 2.17i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.807 + 4.58i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 4.88T + 53T^{2} \)
59 \( 1 + (-5.52 - 2.01i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.68 - 9.56i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.08 + 7.62i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (7.24 + 12.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.219 - 0.380i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.88 - 2.42i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.86 - 5.76i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (1.23 - 2.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.34 + 3.03i)T + (74.3 - 62.3i)T^{2} \)
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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

−12.27984765110773933249459842058, −11.64046437076165661701242221787, −10.62097257077476205434893615433, −10.02259754191713248193201229388, −9.166688843380901451584232971081, −8.198235382879587394565269340052, −7.33776927985196859506566506188, −4.96454589532243074318907571291, −3.73214203020485559653196567956, −2.34945125371695113833602594920, 0.25665037559232559196602008594, 2.55736727071600024116134396163, 5.28687603632916609731834816552, 6.43527984245883621894354933076, 7.32873184394174156448956977379, 8.176065264587479289317955179090, 8.656868677983534374710520026823, 10.09140868621805483454265671604, 11.00440053380895066718661299153, 12.27585061433508168867278828369

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