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Properties

Label 2-189-63.16-c3-0-1
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: $\chi_{189} (100, \cdot )$
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

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

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