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Properties

Label 2-189-63.25-c3-0-12
Degree $2$
Conductor $189$
Sign $0.264 + 0.964i$
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  − 3.66·2-s + 5.46·4-s + (7.38 − 12.7i)5-s + (15.9 − 9.46i)7-s + 9.29·8-s + (−27.1 + 46.9i)10-s + (24.2 + 42.0i)11-s + (−33.6 − 58.1i)13-s + (−58.4 + 34.7i)14-s − 77.8·16-s + (5.40 − 9.36i)17-s + (67.2 + 116. i)19-s + (40.3 − 69.9i)20-s + (−89.1 − 154. i)22-s + (42.1 − 73.0i)23-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.683·4-s + (0.660 − 1.14i)5-s + (0.859 − 0.511i)7-s + 0.410·8-s + (−0.857 + 1.48i)10-s + (0.665 + 1.15i)11-s + (−0.716 − 1.24i)13-s + (−1.11 + 0.663i)14-s − 1.21·16-s + (0.0771 − 0.133i)17-s + (0.811 + 1.40i)19-s + (0.451 − 0.782i)20-s + (−0.864 − 1.49i)22-s + (0.382 − 0.662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.264 + 0.964i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.264 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.830919 - 0.633857i\)
\(L(\frac12)\) \(\approx\) \(0.830919 - 0.633857i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.9 + 9.46i)T \)
good2 \( 1 + 3.66T + 8T^{2} \)
5 \( 1 + (-7.38 + 12.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-24.2 - 42.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (33.6 + 58.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-5.40 + 9.36i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-67.2 - 116. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-42.1 + 73.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-55.1 + 95.4i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 151.T + 2.97e4T^{2} \)
37 \( 1 + (152. + 263. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (127. + 220. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-41.3 + 71.5i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 46.0T + 1.03e5T^{2} \)
53 \( 1 + (-3.20 + 5.55i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 11.1T + 2.05e5T^{2} \)
61 \( 1 - 272.T + 2.26e5T^{2} \)
67 \( 1 + 57.4T + 3.00e5T^{2} \)
71 \( 1 + 521.T + 3.57e5T^{2} \)
73 \( 1 + (189. - 327. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 944.T + 4.93e5T^{2} \)
83 \( 1 + (-411. + 711. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-12.4 - 21.6i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-22.7 + 39.3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.87669142479575731213705797669, −10.35015917736120833514548519093, −9.933138194947564018361855629665, −8.919966016313458853025168879256, −8.025144359517433884250371445776, −7.21807107770300978399557906678, −5.39845590524288274303718915093, −4.43118783859940601604727646210, −1.85180784128855774711182601127, −0.822389868329152742026628958203, 1.41429294144255017524179765329, 2.80622989780823363763796643853, 4.87092918469767544117685085186, 6.44947404361512003396951883468, 7.25481355764716864253360927651, 8.543394896940304221535954648112, 9.267752034814151315557260288592, 10.16747364080807543368838758253, 11.29136554064528540325006100108, 11.61108067004376515978998839737

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