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Properties

Label 2-189-63.59-c3-0-18
Degree $2$
Conductor $189$
Sign $0.100 + 0.994i$
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  + (3.00 + 1.73i)2-s + (2.00 + 3.47i)4-s − 18.0·5-s + (12.2 − 13.9i)7-s − 13.8i·8-s + (−54.1 − 31.2i)10-s − 11.7i·11-s + (−45.9 − 26.5i)13-s + (60.7 − 20.6i)14-s + (39.9 − 69.2i)16-s + (5.93 − 10.2i)17-s + (−12.1 + 7.04i)19-s + (−36.2 − 62.7i)20-s + (20.4 − 35.4i)22-s − 161. i·23-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.434i)4-s − 1.61·5-s + (0.659 − 0.751i)7-s − 0.610i·8-s + (−1.71 − 0.989i)10-s − 0.323i·11-s + (−0.980 − 0.566i)13-s + (1.16 − 0.394i)14-s + (0.624 − 1.08i)16-s + (0.0847 − 0.146i)17-s + (−0.147 + 0.0850i)19-s + (−0.405 − 0.701i)20-s + (0.198 − 0.343i)22-s − 1.46i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.100 + 0.994i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.100 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.13072 - 1.02273i\)
\(L(\frac12)\) \(\approx\) \(1.13072 - 1.02273i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.2 + 13.9i)T \)
good2 \( 1 + (-3.00 - 1.73i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 18.0T + 125T^{2} \)
11 \( 1 + 11.7iT - 1.33e3T^{2} \)
13 \( 1 + (45.9 + 26.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-5.93 + 10.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (12.1 - 7.04i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 161. iT - 1.21e4T^{2} \)
29 \( 1 + (131. - 75.8i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (78.0 - 45.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-185. - 320. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (129. - 223. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (173. + 300. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-35.8 + 62.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-143. - 82.9i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (287. + 498. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-429. - 248. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-350. - 606. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.13e3iT - 3.57e5T^{2} \)
73 \( 1 + (-281. - 162. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-261. + 453. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-410. - 710. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (181. + 313. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (273. - 157. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.10539659583599072060421123828, −11.15317147484158398779154902648, −10.10929668354121607879026195963, −8.374449359821441258685335407142, −7.55199938928234493966455952259, −6.74458743734715741322971808586, −5.10508171491592700219622410166, −4.36547808787801248232770660047, −3.33912514898643260148581894431, −0.47519642508463082422501121689, 2.17956505276857605070783344612, 3.63145088431378033624617973694, 4.50312909437765548202401722717, 5.49609848983036088058697374400, 7.36614248030261613664195106218, 8.111661917558994791543542604932, 9.330210673266795128484214827630, 11.06197132640056922594353150561, 11.61008740307741345530341858030, 12.18852791409364998561693027395

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