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Properties

Label 2-189-189.184-c1-0-21
Degree $2$
Conductor $189$
Sign $0.475 - 0.879i$
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  + (−0.452 − 2.56i)2-s + (−1.35 − 1.07i)3-s + (−4.49 + 1.63i)4-s + (−0.462 + 2.62i)5-s + (−2.15 + 3.96i)6-s + (−1.33 − 2.28i)7-s + (3.63 + 6.28i)8-s + (0.668 + 2.92i)9-s + 6.94·10-s + (0.194 + 1.10i)11-s + (7.86 + 2.64i)12-s + (−3.00 − 2.52i)13-s + (−5.25 + 4.46i)14-s + (3.45 − 3.05i)15-s + (7.15 − 6.00i)16-s − 7.84·17-s + ⋯
L(s)  = 1  + (−0.319 − 1.81i)2-s + (−0.781 − 0.623i)3-s + (−2.24 + 0.818i)4-s + (−0.206 + 1.17i)5-s + (−0.880 + 1.61i)6-s + (−0.504 − 0.863i)7-s + (1.28 + 2.22i)8-s + (0.222 + 0.974i)9-s + 2.19·10-s + (0.0585 + 0.332i)11-s + (2.26 + 0.762i)12-s + (−0.834 − 0.700i)13-s + (−1.40 + 1.19i)14-s + (0.893 − 0.788i)15-s + (1.78 − 1.50i)16-s − 1.90·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.475 - 0.879i$
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: \((2,\ 189,\ (\ :1/2),\ 0.475 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0408291 + 0.0243336i\)
\(L(\frac12)\) \(\approx\) \(0.0408291 + 0.0243336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.35 + 1.07i)T \)
7 \( 1 + (1.33 + 2.28i)T \)
good2 \( 1 + (0.452 + 2.56i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.462 - 2.62i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.194 - 1.10i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (3.00 + 2.52i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 7.84T + 17T^{2} \)
19 \( 1 + 0.855T + 19T^{2} \)
23 \( 1 + (-3.33 - 2.80i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.53 - 2.96i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.758 - 0.275i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (4.18 + 7.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.54 + 1.29i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.49 + 1.99i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.968 - 0.352i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.191 + 0.331i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.12 + 4.30i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (4.06 + 1.47i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.63 + 9.26i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.64 + 6.32i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.74 - 3.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.716 + 4.06i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.86 - 3.24i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 7.13T + 89T^{2} \)
97 \( 1 + (-12.8 - 4.67i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.49102832958022740979438203221, −10.77370241238829066328107750563, −10.41283726920063369130715676572, −9.197695105611344800987920046785, −7.56699525137460804608210918314, −6.74120998801283189426733745291, −4.81417232558763090812818901129, −3.43601693333834361079098443546, −2.13229785140601703809104758404, −0.05000077147825499441608221617, 4.44208391705881073112133498940, 5.03252854985647985858174360237, 6.14975380509979097241268400172, 6.93840767203598758681162534545, 8.640907307983239205242481306147, 8.963063499891014747450532978118, 9.879573252666095253223481122880, 11.49316646698638804570338166159, 12.60853225692503658039006920194, 13.43995633126600257848632807146

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