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Properties

Label 2-189-189.184-c1-0-2
Degree $2$
Conductor $189$
Sign $0.0422 - 0.999i$
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  + (−0.0572 − 0.324i)2-s + (−0.758 + 1.55i)3-s + (1.77 − 0.646i)4-s + (−0.607 + 3.44i)5-s + (0.548 + 0.157i)6-s + (−2.50 + 0.842i)7-s + (−0.641 − 1.11i)8-s + (−1.84 − 2.36i)9-s + 1.15·10-s + (0.460 + 2.61i)11-s + (−0.341 + 3.25i)12-s + (0.586 + 0.491i)13-s + (0.416 + 0.765i)14-s + (−4.90 − 3.56i)15-s + (2.57 − 2.15i)16-s − 1.81·17-s + ⋯
L(s)  = 1  + (−0.0404 − 0.229i)2-s + (−0.438 + 0.898i)3-s + (0.888 − 0.323i)4-s + (−0.271 + 1.54i)5-s + (0.224 + 0.0641i)6-s + (−0.947 + 0.318i)7-s + (−0.226 − 0.392i)8-s + (−0.616 − 0.787i)9-s + 0.364·10-s + (0.138 + 0.788i)11-s + (−0.0985 + 0.940i)12-s + (0.162 + 0.136i)13-s + (0.111 + 0.204i)14-s + (−1.26 − 0.919i)15-s + (0.643 − 0.539i)16-s − 0.439·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.0422 - 0.999i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.0422 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.741205 + 0.710520i\)
\(L(\frac12)\) \(\approx\) \(0.741205 + 0.710520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.758 - 1.55i)T \)
7 \( 1 + (2.50 - 0.842i)T \)
good2 \( 1 + (0.0572 + 0.324i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.607 - 3.44i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.460 - 2.61i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.586 - 0.491i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 1.81T + 17T^{2} \)
19 \( 1 - 4.97T + 19T^{2} \)
23 \( 1 + (-6.18 - 5.19i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.32 + 2.79i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.527 + 0.191i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (4.72 + 8.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.66 + 2.23i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.12 - 1.49i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.170 + 0.0621i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.656 - 1.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.2 + 9.46i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-9.86 - 3.58i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.08 - 11.8i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.38 + 4.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.98 + 5.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.89 - 16.4i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.412 - 0.345i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 5.76T + 89T^{2} \)
97 \( 1 + (9.14 + 3.32i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.39765186153128722403613819770, −11.50191361752778941135920198769, −10.89205131174345841611707907893, −10.01609309161682998277726730972, −9.355717669318102618868693694878, −7.23210098794592928862528050118, −6.63696812109804897487189645505, −5.55110026412446612694928039982, −3.63143170709724222986747174176, −2.73900358064909771381853565607, 1.02782482924729066134235022797, 3.09633597038382702465864724749, 5.02605879970104938386486126202, 6.19946777938386519259596488681, 7.07343043232026063486911974360, 8.203139218720853648443122824808, 8.959079484825854403265252846001, 10.61169015545292027384407191515, 11.65938362987356430766729844479, 12.37604133696437731251599292366

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