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Properties

Label 2-189-63.47-c3-0-15
Degree $2$
Conductor $189$
Sign $-0.658 + 0.752i$
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  + (0.223 − 0.128i)2-s + (−3.96 + 6.87i)4-s − 6.38·5-s + (−1.54 + 18.4i)7-s + 4.10i·8-s + (−1.42 + 0.822i)10-s − 61.0i·11-s + (8.78 − 5.07i)13-s + (2.03 + 4.31i)14-s + (−31.2 − 54.0i)16-s + (−22.5 − 38.9i)17-s + (−69.6 − 40.2i)19-s + (25.3 − 43.8i)20-s + (−7.86 − 13.6i)22-s + 27.6i·23-s + ⋯
L(s)  = 1  + (0.0788 − 0.0455i)2-s + (−0.495 + 0.858i)4-s − 0.571·5-s + (−0.0833 + 0.996i)7-s + 0.181i·8-s + (−0.0450 + 0.0260i)10-s − 1.67i·11-s + (0.187 − 0.108i)13-s + (0.0388 + 0.0824i)14-s + (−0.487 − 0.844i)16-s + (−0.321 − 0.556i)17-s + (−0.840 − 0.485i)19-s + (0.283 − 0.490i)20-s + (−0.0762 − 0.132i)22-s + 0.250i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0933162 - 0.205779i\)
\(L(\frac12)\) \(\approx\) \(0.0933162 - 0.205779i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.54 - 18.4i)T \)
good2 \( 1 + (-0.223 + 0.128i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 6.38T + 125T^{2} \)
11 \( 1 + 61.0iT - 1.33e3T^{2} \)
13 \( 1 + (-8.78 + 5.07i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (22.5 + 38.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.6 + 40.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 27.6iT - 1.21e4T^{2} \)
29 \( 1 + (48.9 + 28.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-92.1 - 53.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-95.6 + 165. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-15.0 - 26.1i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (185. - 320. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (248. + 429. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (601. - 347. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-317. + 550. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-647. + 373. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (82.3 - 142. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 278. iT - 3.57e5T^{2} \)
73 \( 1 + (313. - 181. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (278. + 482. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (514. - 891. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (730. - 1.26e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-878. - 507. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.63579804376770982218580302091, −11.22775968629178438750304616572, −9.475910326208812366743958831791, −8.545269563349721998076400627515, −8.014970477474222860804226007272, −6.47340259662196967204811277839, −5.20343893123341573090657853532, −3.81289320344306554086246898299, −2.75671689566477365685347886691, −0.097563767022099897397784867819, 1.67222474031562838019755138583, 4.02325918243298047565730739236, 4.66019537454402443176304166841, 6.25533993950910455980193775462, 7.26499908692483239163518196797, 8.413626653756766386583345654864, 9.783878468436254464522460773916, 10.29538669989512209961930413910, 11.37944446415885423935679926699, 12.64681591183354330788608600514

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