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Properties

Label 2-189-63.41-c1-0-2
Degree $2$
Conductor $189$
Sign $0.994 + 0.101i$
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  + (0.555 − 0.320i)2-s + (−0.794 + 1.37i)4-s + (1.10 − 1.91i)5-s + (2.60 + 0.458i)7-s + 2.30i·8-s − 1.41i·10-s + (2.93 − 1.69i)11-s + (1.56 + 0.901i)13-s + (1.59 − 0.581i)14-s + (−0.849 − 1.47i)16-s − 5.96·17-s + 1.64i·19-s + (1.75 + 3.04i)20-s + (1.08 − 1.88i)22-s + (−2.05 − 1.18i)23-s + ⋯
L(s)  = 1  + (0.392 − 0.226i)2-s + (−0.397 + 0.687i)4-s + (0.494 − 0.856i)5-s + (0.984 + 0.173i)7-s + 0.813i·8-s − 0.448i·10-s + (0.885 − 0.511i)11-s + (0.432 + 0.249i)13-s + (0.426 − 0.155i)14-s + (−0.212 − 0.367i)16-s − 1.44·17-s + 0.377i·19-s + (0.392 + 0.680i)20-s + (0.232 − 0.401i)22-s + (−0.428 − 0.247i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.994 + 0.101i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.994 + 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50500 - 0.0761991i\)
\(L(\frac12)\) \(\approx\) \(1.50500 - 0.0761991i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.458i)T \)
good2 \( 1 + (-0.555 + 0.320i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.10 + 1.91i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.93 + 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.56 - 0.901i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 - 1.64iT - 19T^{2} \)
23 \( 1 + (2.05 + 1.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (9.28 + 5.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + (0.455 - 0.788i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.96 + 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.123 - 0.213i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.87iT - 53T^{2} \)
59 \( 1 + (5.39 - 9.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.22 - 0.709i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.99 + 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 0.426iT - 73T^{2} \)
79 \( 1 + (-2.49 - 4.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.28 + 7.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (-6.30 + 3.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.62587358770088276914817300987, −11.66206705624164096864415814433, −10.96005820912406186924493199304, −9.039540025726596457443630239269, −8.867699523711137581088854198916, −7.62676061199001084120187581741, −5.97425901353944102708219865840, −4.81146582762169884211025334570, −3.87575815086337407193920184575, −1.90316335246636950398364869551, 1.84994846296907642085155337872, 3.95678900458276712235893860172, 5.08044589968574192994632248007, 6.30786315295730248865676933172, 7.10238700584501184658462893689, 8.686737594234473391528472471615, 9.664538595223715725371900061405, 10.72488760230915338414251103475, 11.35687866283116089849341410560, 12.85952223929554062246987584336

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