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Properties

Label 2-189-63.4-c3-0-11
Degree $2$
Conductor $189$
Sign $0.667 + 0.744i$
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  + (−1.18 + 2.06i)2-s + (1.17 + 2.02i)4-s − 18.4·5-s + (−16.1 + 8.97i)7-s − 24.5·8-s + (21.9 − 38.0i)10-s + 56.7·11-s + (3.61 − 6.26i)13-s + (0.776 − 44.0i)14-s + (19.8 − 34.4i)16-s + (42.2 − 73.2i)17-s + (−1.77 − 3.07i)19-s + (−21.6 − 37.4i)20-s + (−67.5 + 116. i)22-s − 90.6·23-s + ⋯
L(s)  = 1  + (−0.420 + 0.728i)2-s + (0.146 + 0.253i)4-s − 1.65·5-s + (−0.874 + 0.484i)7-s − 1.08·8-s + (0.694 − 1.20i)10-s + 1.55·11-s + (0.0772 − 0.133i)13-s + (0.0148 − 0.840i)14-s + (0.310 − 0.538i)16-s + (0.603 − 1.04i)17-s + (−0.0214 − 0.0371i)19-s + (−0.241 − 0.418i)20-s + (−0.654 + 1.13i)22-s − 0.822·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.343646 - 0.153508i\)
\(L(\frac12)\) \(\approx\) \(0.343646 - 0.153508i\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.1 - 8.97i)T \)
good2 \( 1 + (1.18 - 2.06i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 18.4T + 125T^{2} \)
11 \( 1 - 56.7T + 1.33e3T^{2} \)
13 \( 1 + (-3.61 + 6.26i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-42.2 + 73.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (1.77 + 3.07i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 90.6T + 1.21e4T^{2} \)
29 \( 1 + (25.6 + 44.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (78.2 + 135. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-28.9 - 50.1i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-79.8 + 138. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (20.6 + 35.7i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-72.4 + 125. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (369. - 639. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (112. + 195. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-133. + 231. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (447. + 774. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 52.1T + 3.57e5T^{2} \)
73 \( 1 + (289. - 501. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (247. - 429. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (380. + 658. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (364. + 631. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (565. + 979. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.96881933418475189197198233121, −11.36820589290398252367785549458, −9.576810899702717040950082931343, −8.800218899759770478078480652331, −7.76606532269482957593703207061, −7.01722275215478727085191600142, −6.00132999857194363630040775479, −4.06457764337806945719970200984, −3.13656512179055145144246418938, −0.21767916934221982577307980134, 1.22070831710253072965143298295, 3.36614820255971378585557158673, 4.06150248803047461417875322415, 6.15340001980601264192553919840, 7.10304186944412878526964132110, 8.380041385736030318027227311268, 9.386406401287492348670879176227, 10.39536817632337144636989186636, 11.31155517200047799140117566879, 12.02750270700307238345411705921

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