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Properties

Label 2-189-63.23-c2-0-6
Degree $2$
Conductor $189$
Sign $0.769 + 0.638i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.29i·2-s + 2.33·4-s + (−4.18 + 2.41i)5-s + (6.74 + 1.88i)7-s − 8.17i·8-s + (3.12 + 5.40i)10-s + (11.8 + 6.84i)11-s + (8.59 − 14.8i)13-s + (2.43 − 8.70i)14-s − 1.21·16-s + (7.11 − 4.10i)17-s + (−6.93 + 12.0i)19-s + (−9.77 + 5.64i)20-s + (8.83 − 15.2i)22-s + (16.0 − 9.26i)23-s + ⋯
L(s)  = 1  − 0.645i·2-s + 0.583·4-s + (−0.837 + 0.483i)5-s + (0.963 + 0.268i)7-s − 1.02i·8-s + (0.312 + 0.540i)10-s + (1.07 + 0.621i)11-s + (0.661 − 1.14i)13-s + (0.173 − 0.621i)14-s − 0.0762·16-s + (0.418 − 0.241i)17-s + (−0.364 + 0.631i)19-s + (−0.488 + 0.282i)20-s + (0.401 − 0.695i)22-s + (0.697 − 0.402i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.769 + 0.638i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.769 + 0.638i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.70640 - 0.615509i\)
\(L(\frac12)\) \(\approx\) \(1.70640 - 0.615509i\)
\(L(2)\) 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 \( 1 + (-6.74 - 1.88i)T \)
good2 \( 1 + 1.29iT - 4T^{2} \)
5 \( 1 + (4.18 - 2.41i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-11.8 - 6.84i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.59 + 14.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-7.11 + 4.10i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.93 - 12.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.0 + 9.26i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-18.1 + 10.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 51.5T + 961T^{2} \)
37 \( 1 + (-13.8 + 23.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (18.4 + 10.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-18.1 - 31.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 59.3iT - 2.20e3T^{2} \)
53 \( 1 + (81.0 - 46.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 1.67iT - 3.48e3T^{2} \)
61 \( 1 + 65.4T + 3.72e3T^{2} \)
67 \( 1 + 55.2T + 4.48e3T^{2} \)
71 \( 1 + 14.2iT - 5.04e3T^{2} \)
73 \( 1 + (-32.6 - 56.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + 6.70T + 6.24e3T^{2} \)
83 \( 1 + (11.6 - 6.69i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (13.7 + 7.92i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (40.7 + 70.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.10336614789376425083537877721, −11.13863931386029434417911024919, −10.74518307312725387948875021682, −9.383180452633054207659046788554, −8.010515062943736891852767468769, −7.22953725737261995960084567449, −5.93669332958714452156912904493, −4.22432039070020157679582857175, −3.08628865243745635192915330359, −1.44179033007265591003936445629, 1.52672656387301202112670668648, 3.74052008244906923588106582543, 4.95284196105996536370883596615, 6.34770594749217944481200233913, 7.28246753466207506097444798302, 8.324944490615459306055714154377, 8.990676396040846046379389298546, 10.89511062623835433155978075724, 11.45026080381382172679916627498, 12.13485122506081199214549147787

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