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Properties

Label 2-189-63.41-c3-0-20
Degree $2$
Conductor $189$
Sign $-0.404 + 0.914i$
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 + (−4.13 − 18.0i)7-s + 6.12i·8-s − 75.6i·10-s + (−31.1 + 17.9i)11-s + (6.59 + 3.81i)13-s + (−47.8 − 51.4i)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 + (−0.223 − 0.974i)7-s + 0.270i·8-s − 2.39i·10-s + (−0.854 + 0.493i)11-s + (0.140 + 0.0812i)13-s + (−0.912 − 0.982i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.404 + 0.914i$
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: \((2,\ 189,\ (\ :3/2),\ -0.404 + 0.914i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.81402 - 2.78563i\)
\(L(\frac12)\) \(\approx\) \(1.81402 - 2.78563i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.13 + 18.0i)T \)
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

−12.20187642284064766803193667411, −10.97924546235646965586339482861, −9.989108263530962332561827771577, −8.964764777446698860182113551289, −7.70552382172699280055123209043, −6.02762875459375067655821339570, −4.93467452973695223886885156513, −4.34244248575583294706765298367, −2.62801766334112060015171380899, −1.09618776528919966349590656820, 2.50745744156438582311771240605, 3.47101231914391535532034259497, 5.34557744050961003734821908745, 5.99711896074545145606539578776, 6.75768185995901756556321586742, 8.030141548231037882877345245899, 9.676687390062153679957605359505, 10.41681100017949686520753830363, 11.60875760944349389523809627568, 12.80857294847592698628054290388

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