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Properties

Label 2-189-63.4-c3-0-13
Degree $2$
Conductor $189$
Sign $0.496 + 0.868i$
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  + (1.33 − 2.30i)2-s + (0.454 + 0.787i)4-s − 19.2·5-s + (18.0 + 4.02i)7-s + 23.7·8-s + (−25.6 + 44.3i)10-s + 38.5·11-s + (38.2 − 66.3i)13-s + (33.3 − 36.3i)14-s + (27.9 − 48.4i)16-s + (11.9 − 20.6i)17-s + (−23.6 − 41.0i)19-s + (−8.74 − 15.1i)20-s + (51.3 − 88.8i)22-s + 13.5·23-s + ⋯
L(s)  = 1  + (0.470 − 0.815i)2-s + (0.0568 + 0.0983i)4-s − 1.72·5-s + (0.976 + 0.217i)7-s + 1.04·8-s + (−0.810 + 1.40i)10-s + 1.05·11-s + (0.816 − 1.41i)13-s + (0.636 − 0.693i)14-s + (0.436 − 0.756i)16-s + (0.170 − 0.294i)17-s + (−0.285 − 0.495i)19-s + (−0.0977 − 0.169i)20-s + (0.497 − 0.861i)22-s + 0.122·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.496 + 0.868i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.496 + 0.868i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.90542 - 1.10519i\)
\(L(\frac12)\) \(\approx\) \(1.90542 - 1.10519i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.0 - 4.02i)T \)
good2 \( 1 + (-1.33 + 2.30i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 19.2T + 125T^{2} \)
11 \( 1 - 38.5T + 1.33e3T^{2} \)
13 \( 1 + (-38.2 + 66.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-11.9 + 20.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (23.6 + 41.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 13.5T + 1.21e4T^{2} \)
29 \( 1 + (-53.5 - 92.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-79.3 - 137. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-23.0 - 39.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (101. - 175. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (42.1 + 72.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-236. + 410. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-39.8 + 69.0i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (158. + 274. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-81.8 + 141. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (270. + 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 810.T + 3.57e5T^{2} \)
73 \( 1 + (142. - 246. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (367. - 635. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-290. - 503. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-463. - 803. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (413. + 715. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.90135378291344068682777671729, −11.23375294872529625131590543212, −10.56372491989880926891104218948, −8.613334251873932802392904977842, −8.000908723874636556748589021140, −6.97063051616382190581155907910, −4.99789491602026039685993041068, −3.98534242068839504828111491986, −3.08146690813583604554427925177, −1.09406710917779886894336799329, 1.35769615766628928230951025282, 4.08365445915050875040818903733, 4.38296358509672183870141820507, 6.10170374930495452279661405660, 7.12744788748079851200373915511, 7.935641091196380473827807930937, 8.870430738842660304557877519056, 10.63279326828524049004131392654, 11.50060322610509086680798001041, 11.98158781221065596080853945750

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