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Properties

Label 2-189-63.16-c3-0-0
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: $\chi_{189} (100, \cdot )$
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

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

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