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Properties

Label 2-189-63.41-c3-0-6
Degree $2$
Conductor $189$
Sign $-0.162 - 0.986i$
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  + (2.31 − 1.33i)2-s + (−0.417 + 0.723i)4-s + (0.223 − 0.386i)5-s + (−7.50 + 16.9i)7-s + 23.6i·8-s − 1.19i·10-s + (−34.2 + 19.7i)11-s + (−68.4 − 39.5i)13-s + (5.25 + 49.2i)14-s + (28.3 + 49.0i)16-s + 9.74·17-s + 73.1i·19-s + (0.186 + 0.323i)20-s + (−52.9 + 91.7i)22-s + (126. + 73.0i)23-s + ⋯
L(s)  = 1  + (0.819 − 0.473i)2-s + (−0.0522 + 0.0904i)4-s + (0.0199 − 0.0345i)5-s + (−0.405 + 0.914i)7-s + 1.04i·8-s − 0.0377i·10-s + (−0.939 + 0.542i)11-s + (−1.46 − 0.843i)13-s + (0.100 + 0.941i)14-s + (0.442 + 0.766i)16-s + 0.139·17-s + 0.882i·19-s + (0.00208 + 0.00361i)20-s + (−0.513 + 0.889i)22-s + (1.14 + 0.661i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.162 - 0.986i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.162 - 0.986i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.970659 + 1.14402i\)
\(L(\frac12)\) \(\approx\) \(0.970659 + 1.14402i\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.50 - 16.9i)T \)
good2 \( 1 + (-2.31 + 1.33i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-0.223 + 0.386i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (34.2 - 19.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (68.4 + 39.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 9.74T + 4.91e3T^{2} \)
19 \( 1 - 73.1iT - 6.85e3T^{2} \)
23 \( 1 + (-126. - 73.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-134. + 77.4i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (9.87 + 5.70i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (-53.3 + 92.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (45.6 + 78.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-276. - 479. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 239. iT - 1.48e5T^{2} \)
59 \( 1 + (-126. + 218. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-342. + 197. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-13.5 + 23.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 - 923. iT - 3.89e5T^{2} \)
79 \( 1 + (-280. - 485. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (281. + 487. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 644.T + 7.04e5T^{2} \)
97 \( 1 + (427. - 246. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.62093710070954263081956094299, −11.77613031634955064287832883953, −10.53212171215274972487601775485, −9.524133338264361595483012358320, −8.306898575033517630603638830594, −7.29021160748161917556272975421, −5.51197148139485529013313069740, −4.96297603539787523368461849047, −3.27483148537651097143122047697, −2.36106640879849258036160951385, 0.48749818331657823518276473116, 2.91843016480549460698218418355, 4.44301374499853205442602436106, 5.19467047914434887120632320144, 6.68900379016023431736966180398, 7.22237700965781075918733132927, 8.830643490920020352457626361777, 10.03046844186423850114730798959, 10.68967128942570686932061592027, 12.17784599740366740353080599930

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