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Properties

Label 2-189-63.4-c3-0-18
Degree $2$
Conductor $189$
Sign $-0.226 + 0.973i$
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  + (1.67 − 2.90i)2-s + (−1.62 − 2.80i)4-s + 8.70·5-s + (−14.4 − 11.6i)7-s + 15.9·8-s + (14.5 − 25.2i)10-s + 11.7·11-s + (26.5 − 46.0i)13-s + (−57.9 + 22.3i)14-s + (39.7 − 68.7i)16-s + (22.5 − 39.0i)17-s + (−13.0 − 22.6i)19-s + (−14.1 − 24.4i)20-s + (19.7 − 34.2i)22-s + 58.1·23-s + ⋯
L(s)  = 1  + (0.592 − 1.02i)2-s + (−0.202 − 0.351i)4-s + 0.778·5-s + (−0.777 − 0.628i)7-s + 0.704·8-s + (0.461 − 0.799i)10-s + 0.323·11-s + (0.567 − 0.982i)13-s + (−1.10 + 0.425i)14-s + (0.620 − 1.07i)16-s + (0.321 − 0.557i)17-s + (−0.158 − 0.273i)19-s + (−0.157 − 0.273i)20-s + (0.191 − 0.331i)22-s + 0.527·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.226 + 0.973i$
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: \((2,\ 189,\ (\ :3/2),\ -0.226 + 0.973i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68991 - 2.12891i\)
\(L(\frac12)\) \(\approx\) \(1.68991 - 2.12891i\)
\(L(\frac{5}{2})\) 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 \( 1 + (14.4 + 11.6i)T \)
good2 \( 1 + (-1.67 + 2.90i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 8.70T + 125T^{2} \)
11 \( 1 - 11.7T + 1.33e3T^{2} \)
13 \( 1 + (-26.5 + 46.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-22.5 + 39.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (13.0 + 22.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 58.1T + 1.21e4T^{2} \)
29 \( 1 + (36.7 + 63.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-157. - 272. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (189. + 327. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (181. - 313. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-58.7 - 101. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-114. + 197. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (307. - 532. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-288. - 499. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (223. - 386. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (148. + 257. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 866.T + 3.57e5T^{2} \)
73 \( 1 + (-283. + 490. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-183. + 317. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-510. - 884. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-247. - 427. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (76.3 + 132. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.97225058885460762332199367751, −10.76119178243338411102112504353, −10.22282814836273868562500843111, −9.184433068829480113175696940991, −7.64383846593624525148903121923, −6.43758966074818814365115062279, −5.14374527549630481690479636178, −3.73518773931237116355135890915, −2.75017079905099374114588867773, −1.11382974102488884727637898498, 1.86046026842402013371482694154, 3.78331626043978478964908613778, 5.23976829827830267158784438703, 6.21422513641308587267691060621, 6.71097336067162657480118653102, 8.212188610390715875040663967977, 9.343747912401843381344033798847, 10.23953992541983665151308065960, 11.55371953341096128266015255335, 12.75018429202057337968897372777

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