Properties

Label 2-189-189.4-c1-0-20
Degree $2$
Conductor $189$
Sign $-0.542 - 0.839i$
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  + (−0.203 − 1.15i)2-s + (−1.51 − 0.846i)3-s + (0.582 − 0.212i)4-s + (−3.41 + 1.24i)5-s + (−0.670 + 1.92i)6-s + (−1.99 + 1.73i)7-s + (−1.53 − 2.66i)8-s + (1.56 + 2.55i)9-s + (2.13 + 3.69i)10-s + (0.000515 + 0.000187i)11-s + (−1.06 − 0.172i)12-s + (−2.69 + 0.981i)13-s + (2.41 + 1.95i)14-s + (6.20 + 1.01i)15-s + (−1.81 + 1.52i)16-s + (−2.53 − 4.39i)17-s + ⋯
L(s)  = 1  + (−0.144 − 0.817i)2-s + (−0.872 − 0.488i)3-s + (0.291 − 0.106i)4-s + (−1.52 + 0.555i)5-s + (−0.273 + 0.784i)6-s + (−0.755 + 0.654i)7-s + (−0.544 − 0.942i)8-s + (0.522 + 0.852i)9-s + (0.674 + 1.16i)10-s + (0.000155 + 5.66e−5i)11-s + (−0.306 − 0.0498i)12-s + (−0.748 + 0.272i)13-s + (0.644 + 0.523i)14-s + (1.60 + 0.260i)15-s + (−0.454 + 0.381i)16-s + (−0.616 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.542 - 0.839i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.542 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0322328 + 0.0592172i\)
\(L(\frac12)\) \(\approx\) \(0.0322328 + 0.0592172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 + 0.846i)T \)
7 \( 1 + (1.99 - 1.73i)T \)
good2 \( 1 + (0.203 + 1.15i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (3.41 - 1.24i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-0.000515 - 0.000187i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (2.69 - 0.981i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.53 + 4.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.08 + 1.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.20 - 6.85i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (7.97 + 2.90i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.25 - 0.456i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + 4.25T + 37T^{2} \)
41 \( 1 + (-3.15 + 1.14i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.87 + 10.6i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.51 - 2.37i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-2.44 + 4.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.33 - 4.47i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.54 + 0.563i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.57 - 14.5i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.45 - 7.71i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.56T + 73T^{2} \)
79 \( 1 + (0.856 + 4.85i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.35 - 1.58i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (0.167 - 0.289i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.55 - 8.80i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.63392096924275449594601642231, −11.44549163728968311279211137687, −10.27553474172455282994019805296, −9.175909897129275939796569282283, −7.33518965388115106603231095575, −7.01574920053930867224803805568, −5.55756962084089851845940839510, −3.82862097056997281550202254587, −2.46449574324364116336215708928, −0.06480004619736974379204160465, 3.60699643606522182720638271508, 4.66577350848300917369531636037, 6.05930051999305485373698136679, 7.07456386354010369889576275411, 7.88646350492782968917484096100, 9.042339875450655229421873516323, 10.46653590915344483639487655975, 11.24563070821640045795037954448, 12.27706766725929458683019221881, 12.75394576588714846441679680077

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