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Properties

Label 2-189-63.4-c3-0-20
Degree $2$
Conductor $189$
Sign $-0.519 - 0.854i$
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  + (2.51 − 4.35i)2-s + (−8.65 − 14.9i)4-s − 0.150·5-s + (−18.0 + 3.93i)7-s − 46.8·8-s + (−0.378 + 0.654i)10-s − 46.9·11-s + (−8.75 + 15.1i)13-s + (−28.3 + 88.7i)14-s + (−48.7 + 84.3i)16-s + (35.7 − 61.9i)17-s + (−57.4 − 99.4i)19-s + (1.30 + 2.25i)20-s + (−118. + 204. i)22-s + 135.·23-s + ⋯
L(s)  = 1  + (0.889 − 1.54i)2-s + (−1.08 − 1.87i)4-s − 0.0134·5-s + (−0.977 + 0.212i)7-s − 2.07·8-s + (−0.0119 + 0.0207i)10-s − 1.28·11-s + (−0.186 + 0.323i)13-s + (−0.542 + 1.69i)14-s + (−0.761 + 1.31i)16-s + (0.509 − 0.883i)17-s + (−0.693 − 1.20i)19-s + (0.0145 + 0.0252i)20-s + (−1.14 + 1.98i)22-s + 1.22·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.519 - 0.854i$
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: \((2,\ 189,\ (\ :3/2),\ -0.519 - 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.585346 + 1.04063i\)
\(L(\frac12)\) \(\approx\) \(0.585346 + 1.04063i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.0 - 3.93i)T \)
good2 \( 1 + (-2.51 + 4.35i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 0.150T + 125T^{2} \)
11 \( 1 + 46.9T + 1.33e3T^{2} \)
13 \( 1 + (8.75 - 15.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-35.7 + 61.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (57.4 + 99.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 135.T + 1.21e4T^{2} \)
29 \( 1 + (-30.1 - 52.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (5.10 + 8.83i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-152. - 263. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-142. + 247. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (234. + 406. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-168. + 292. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (13.1 - 22.7i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (289. + 501. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (403. - 699. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-60.6 - 105. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 26.5T + 3.57e5T^{2} \)
73 \( 1 + (60.9 - 105. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-29.3 + 50.7i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (298. + 517. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (286. + 495. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (335. + 581. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.59585196971506742076107188240, −10.62889498351626273840125967636, −9.862721359795421724957447933238, −8.922026232133240087130847026819, −7.11251610607191142625275621341, −5.61235977916550841054452542651, −4.67193758834601065016965652329, −3.22396731901320396669005773133, −2.39771469537620280846428140528, −0.37460333974996222641584804845, 3.11626782199266994655794615860, 4.32501179296662523437965410057, 5.64403006824404851896768315496, 6.29489382313581472293654454188, 7.57891113548824689673285054242, 8.151775462848761618944312057686, 9.609387384014885252242426763450, 10.72498711540883443125603753759, 12.57983934976084080840797224679, 12.83831712727543315607684519756

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