Properties

Label 2-189-63.4-c3-0-14
Degree $2$
Conductor $189$
Sign $0.635 + 0.772i$
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  + (−2.43 + 4.21i)2-s + (−7.86 − 13.6i)4-s + 6.21·5-s + (−4.64 + 17.9i)7-s + 37.6·8-s + (−15.1 + 26.2i)10-s − 55.5·11-s + (−6.74 + 11.6i)13-s + (−64.3 − 63.2i)14-s + (−28.7 + 49.8i)16-s + (43.1 − 74.7i)17-s + (−42.1 − 72.9i)19-s + (−48.8 − 84.6i)20-s + (135. − 234. i)22-s − 10.5·23-s + ⋯
L(s)  = 1  + (−0.861 + 1.49i)2-s + (−0.983 − 1.70i)4-s + 0.555·5-s + (−0.250 + 0.968i)7-s + 1.66·8-s + (−0.478 + 0.828i)10-s − 1.52·11-s + (−0.143 + 0.249i)13-s + (−1.22 − 1.20i)14-s + (−0.449 + 0.779i)16-s + (0.616 − 1.06i)17-s + (−0.508 − 0.880i)19-s + (−0.546 − 0.945i)20-s + (1.31 − 2.27i)22-s − 0.0955·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.635 + 0.772i$
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: \((2,\ 189,\ (\ :3/2),\ 0.635 + 0.772i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.168298 - 0.0794934i\)
\(L(\frac12)\) \(\approx\) \(0.168298 - 0.0794934i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.64 - 17.9i)T \)
good2 \( 1 + (2.43 - 4.21i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 6.21T + 125T^{2} \)
11 \( 1 + 55.5T + 1.33e3T^{2} \)
13 \( 1 + (6.74 - 11.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-43.1 + 74.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (42.1 + 72.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 10.5T + 1.21e4T^{2} \)
29 \( 1 + (76.2 + 132. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-127. - 220. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (172. + 298. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-57.9 + 100. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-46.7 - 80.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-159. + 276. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-136. + 236. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-175. - 303. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-145. + 252. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (416. + 721. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 771.T + 3.57e5T^{2} \)
73 \( 1 + (140. - 242. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (172. - 299. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-62.9 - 108. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-163. - 282. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-124. - 215. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.01678030053966005238376490041, −10.49809750592364922163385802497, −9.609632043042324743561983769335, −8.853832588397808209605580606924, −7.85887242407420343557152845259, −6.88755312592433811058527112955, −5.71434645246407336990415415479, −5.12287885141299540455841109349, −2.49250414381427277291965406225, −0.10800472214479018477384727608, 1.50144938728638796403093130144, 2.85704669811743090696533076401, 4.10172223566009184079194366183, 5.83489182337424146041494978582, 7.63533450855889577907160082883, 8.368595401319113133631919356450, 9.819215742403758584493046927609, 10.25715730471421737830380638106, 10.87168896308018858379660205233, 12.15979861024174693660020780297

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