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Properties

Label 2-189-63.16-c3-0-13
Degree $2$
Conductor $189$
Sign $0.999 - 0.0356i$
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  + (0.491 + 0.851i)2-s + (3.51 − 6.09i)4-s + 18.7·5-s + (10.0 + 15.5i)7-s + 14.7·8-s + (9.20 + 15.9i)10-s − 23.4·11-s + (−23.8 − 41.3i)13-s + (−8.34 + 16.1i)14-s + (−20.8 − 36.1i)16-s + (47.7 + 82.7i)17-s + (28.4 − 49.3i)19-s + (65.8 − 113. i)20-s + (−11.5 − 20.0i)22-s − 32.6·23-s + ⋯
L(s)  = 1  + (0.173 + 0.301i)2-s + (0.439 − 0.761i)4-s + 1.67·5-s + (0.541 + 0.840i)7-s + 0.653·8-s + (0.291 + 0.504i)10-s − 0.644·11-s + (−0.509 − 0.882i)13-s + (−0.159 + 0.309i)14-s + (−0.325 − 0.564i)16-s + (0.681 + 1.18i)17-s + (0.343 − 0.595i)19-s + (0.735 − 1.27i)20-s + (−0.111 − 0.193i)22-s − 0.295·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0356i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.999 - 0.0356i$
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: \((2,\ 189,\ (\ :3/2),\ 0.999 - 0.0356i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.85060 + 0.0507934i\)
\(L(\frac12)\) \(\approx\) \(2.85060 + 0.0507934i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.0 - 15.5i)T \)
good2 \( 1 + (-0.491 - 0.851i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 18.7T + 125T^{2} \)
11 \( 1 + 23.4T + 1.33e3T^{2} \)
13 \( 1 + (23.8 + 41.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-47.7 - 82.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-28.4 + 49.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 32.6T + 1.21e4T^{2} \)
29 \( 1 + (81.3 - 140. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-20.2 + 35.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (127. - 221. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-35.9 - 62.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-237. + 411. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (132. + 229. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (13.1 + 22.8i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-19.5 + 33.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (210. + 365. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-184. + 319. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 685.T + 3.57e5T^{2} \)
73 \( 1 + (-113. - 196. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-343. - 594. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-49.0 + 85.0i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (529. - 916. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (706. - 1.22e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.26900330729849538959848567554, −10.81108632783355467197619304335, −10.21018654742447273526540922697, −9.309750452303569613878471492823, −8.007296592801679014487976185533, −6.60449228808448171542805239212, −5.45644629982959721353720318247, −5.31629542358770348473122987475, −2.59398726796366708593288561530, −1.55607439227918508577671706664, 1.63870673362846676336421015415, 2.76251388829289116293334016571, 4.43796894393305973576503663952, 5.67635451336164692370907595506, 7.02118831202648465688386196275, 7.84249264234784072871945424862, 9.365616375049358384806348413576, 10.16257327131133438910345094397, 11.11808523668542117892302225525, 12.13589796148278213956814664790

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