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Properties

Label 2-189-63.20-c3-0-19
Degree $2$
Conductor $189$
Sign $-0.179 + 0.983i$
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  + (3.28 + 1.89i)2-s + (3.19 + 5.53i)4-s + (−9.97 − 17.2i)5-s + (−13.5 + 12.6i)7-s − 6.12i·8-s − 75.6i·10-s + (−31.1 − 17.9i)11-s + (−6.59 + 3.81i)13-s + (−68.4 + 15.6i)14-s + (37.1 − 64.3i)16-s − 21.4·17-s − 97.3i·19-s + (63.7 − 110. i)20-s + (−68.2 − 118. i)22-s + (35.6 − 20.6i)23-s + ⋯
L(s)  = 1  + (1.16 + 0.670i)2-s + (0.399 + 0.691i)4-s + (−0.892 − 1.54i)5-s + (−0.732 + 0.680i)7-s − 0.270i·8-s − 2.39i·10-s + (−0.854 − 0.493i)11-s + (−0.140 + 0.0812i)13-s + (−1.30 + 0.299i)14-s + (0.580 − 1.00i)16-s − 0.306·17-s − 1.17i·19-s + (0.712 − 1.23i)20-s + (−0.661 − 1.14i)22-s + (0.323 − 0.186i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.179 + 0.983i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.179 + 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.920207 - 1.10344i\)
\(L(\frac12)\) \(\approx\) \(0.920207 - 1.10344i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.5 - 12.6i)T \)
good2 \( 1 + (-3.28 - 1.89i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (9.97 + 17.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (31.1 + 17.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (6.59 - 3.81i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 21.4T + 4.91e3T^{2} \)
19 \( 1 + 97.3iT - 6.85e3T^{2} \)
23 \( 1 + (-35.6 + 20.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-51.8 - 29.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (61.1 - 35.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 355.T + 5.06e4T^{2} \)
41 \( 1 + (-7.30 - 12.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (48.8 - 84.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (234. - 406. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 710. iT - 1.48e5T^{2} \)
59 \( 1 + (-232. - 403. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (542. + 313. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (262. + 454. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 115. iT - 3.57e5T^{2} \)
73 \( 1 + 708. iT - 3.89e5T^{2} \)
79 \( 1 + (-128. + 222. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-200. + 347. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 977.T + 7.04e5T^{2} \)
97 \( 1 + (-521. - 300. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.30637340399951916229122661462, −11.29050376686017322479751254849, −9.561337507419276746302037911777, −8.672041369516663532432202868415, −7.60657869759203438276638416598, −6.28694136703765649384870641071, −5.17577812479841111895258503409, −4.52338668412841556228126701645, −3.12318682328642570166513508394, −0.41841083420128471854880521841, 2.55520244495347580403888536515, 3.46118678784681828461872062460, 4.34724065639154293825209617927, 5.96656782954756095173185102783, 7.13195616213782317305391498857, 7.980099544870439210119717587146, 10.01983402548124361641021775111, 10.66934308249388642001845310606, 11.47411195816220679625757176347, 12.41828682346719137205664271789

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