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Properties

Label 2-189-63.59-c3-0-15
Degree $2$
Conductor $189$
Sign $0.666 + 0.745i$
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  + (1.59 + 0.919i)2-s + (−2.31 − 4.00i)4-s − 0.414·5-s + (12.7 + 13.4i)7-s − 23.1i·8-s + (−0.659 − 0.381i)10-s − 49.6i·11-s + (1.43 + 0.828i)13-s + (7.94 + 33.1i)14-s + (2.83 − 4.91i)16-s + (20.6 − 35.7i)17-s + (130. − 75.5i)19-s + (0.957 + 1.65i)20-s + (45.5 − 78.9i)22-s − 131. i·23-s + ⋯
L(s)  = 1  + (0.562 + 0.324i)2-s + (−0.288 − 0.500i)4-s − 0.0370·5-s + (0.688 + 0.725i)7-s − 1.02i·8-s + (−0.0208 − 0.0120i)10-s − 1.35i·11-s + (0.0306 + 0.0176i)13-s + (0.151 + 0.631i)14-s + (0.0443 − 0.0767i)16-s + (0.294 − 0.510i)17-s + (1.57 − 0.911i)19-s + (0.0107 + 0.0185i)20-s + (0.441 − 0.765i)22-s − 1.19i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.666 + 0.745i$
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: \((2,\ 189,\ (\ :3/2),\ 0.666 + 0.745i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.96611 - 0.879645i\)
\(L(\frac12)\) \(\approx\) \(1.96611 - 0.879645i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.7 - 13.4i)T \)
good2 \( 1 + (-1.59 - 0.919i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 0.414T + 125T^{2} \)
11 \( 1 + 49.6iT - 1.33e3T^{2} \)
13 \( 1 + (-1.43 - 0.828i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-20.6 + 35.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-130. + 75.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 131. iT - 1.21e4T^{2} \)
29 \( 1 + (-136. + 78.8i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (15.9 - 9.20i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-173. - 301. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-16.9 + 29.4i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-29.5 - 51.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (108. - 187. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (174. + 100. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (149. + 259. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (69.6 + 40.2i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-128. - 223. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 + (711. + 410. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (52.6 - 91.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (245. + 424. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-754. - 1.30e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.08e3 + 627. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.92004211269285708951737778449, −11.20930386127526308694040744520, −9.921818936141308097511938728253, −8.954338669467708715810672952516, −7.910577031504140424681389987615, −6.42321070405196148283982447433, −5.51649670826330576532534512312, −4.61348852760889169260901114421, −2.98608910322894479791532335346, −0.883970472352140066971117476589, 1.71681308126446531138091097449, 3.49222249271493831223368055935, 4.46443526524716313878656138602, 5.53194176837899499757537563069, 7.43232361131315328493370662594, 7.86874046180075991159438884973, 9.361597743283867183757328334132, 10.35921244133574439502616912360, 11.59157243430632967038056315823, 12.18446369080653823146878004715

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