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Properties

Label 2-189-63.47-c3-0-4
Degree $2$
Conductor $189$
Sign $-0.782 - 0.622i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.68 + 2.12i)2-s + (5.03 − 8.72i)4-s + 2.97·5-s + (−4.05 + 18.0i)7-s + 8.83i·8-s + (−10.9 + 6.32i)10-s − 33.6i·11-s + (22.0 − 12.7i)13-s + (−23.5 − 75.1i)14-s + (21.5 + 37.2i)16-s + (55.6 + 96.3i)17-s + (66.6 + 38.4i)19-s + (14.9 − 25.9i)20-s + (71.5 + 123. i)22-s + 27.8i·23-s + ⋯
L(s)  = 1  + (−1.30 + 0.751i)2-s + (0.629 − 1.09i)4-s + 0.266·5-s + (−0.218 + 0.975i)7-s + 0.390i·8-s + (−0.346 + 0.200i)10-s − 0.922i·11-s + (0.471 − 0.272i)13-s + (−0.448 − 1.43i)14-s + (0.336 + 0.582i)16-s + (0.793 + 1.37i)17-s + (0.804 + 0.464i)19-s + (0.167 − 0.290i)20-s + (0.693 + 1.20i)22-s + 0.252i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.782 - 0.622i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.782 - 0.622i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.225283 + 0.644677i\)
\(L(\frac12)\) \(\approx\) \(0.225283 + 0.644677i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.05 - 18.0i)T \)
good2 \( 1 + (3.68 - 2.12i)T + (4 - 6.92i)T^{2} \)
5 \( 1 - 2.97T + 125T^{2} \)
11 \( 1 + 33.6iT - 1.33e3T^{2} \)
13 \( 1 + (-22.0 + 12.7i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-55.6 - 96.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-66.6 - 38.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 27.8iT - 1.21e4T^{2} \)
29 \( 1 + (87.6 + 50.6i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (57.9 + 33.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (146. - 254. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-62.2 - 107. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (151. - 262. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-183. - 317. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (281. - 162. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (388. - 673. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-162. + 93.6i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-358. + 621. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 762. iT - 3.57e5T^{2} \)
73 \( 1 + (409. - 236. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-353. - 612. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-406. + 703. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-393. + 681. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.51e3 - 871. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.43001427188892072166106228856, −11.24884887740293082563877434335, −10.13223397452286754814331414955, −9.357958022996258244749674724380, −8.394397254792816754227347272589, −7.76291295890902604699997236064, −6.16242584291467376233654901871, −5.75421224840047923219451345649, −3.42134550054572609582649359459, −1.40737574855127053208053602248, 0.50059523214993855113654860189, 1.91191339351660119430396086694, 3.49288093594555035848443408729, 5.19115064921273544111916055469, 7.04523158568890582497135358719, 7.68262907115480107709794109786, 9.116208062531178518556580292648, 9.730029834540250386851364279929, 10.52158600944712585807738451611, 11.45939785800228369641093214342

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