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Properties

Label 2-189-63.47-c3-0-2
Degree $2$
Conductor $189$
Sign $-0.0372 - 0.999i$
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  + (2.65 − 1.53i)2-s + (0.685 − 1.18i)4-s − 5.50·5-s + (−18.4 + 1.02i)7-s + 20.2i·8-s + (−14.5 + 8.42i)10-s + 59.4i·11-s + (12.7 − 7.35i)13-s + (−47.4 + 31.0i)14-s + (36.5 + 63.2i)16-s + (29.3 + 50.7i)17-s + (−66.1 − 38.2i)19-s + (−3.77 + 6.53i)20-s + (90.9 + 157. i)22-s + 26.0i·23-s + ⋯
L(s)  = 1  + (0.937 − 0.541i)2-s + (0.0857 − 0.148i)4-s − 0.492·5-s + (−0.998 + 0.0556i)7-s + 0.896i·8-s + (−0.461 + 0.266i)10-s + 1.62i·11-s + (0.271 − 0.157i)13-s + (−0.905 + 0.592i)14-s + (0.571 + 0.989i)16-s + (0.418 + 0.724i)17-s + (−0.798 − 0.461i)19-s + (−0.0421 + 0.0730i)20-s + (0.881 + 1.52i)22-s + 0.236i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0372 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0372 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.0372 - 0.999i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.0372 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.996705 + 1.03456i\)
\(L(\frac12)\) \(\approx\) \(0.996705 + 1.03456i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 - 1.02i)T \)
good2 \( 1 + (-2.65 + 1.53i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 5.50T + 125T^{2} \)
11 \( 1 - 59.4iT - 1.33e3T^{2} \)
13 \( 1 + (-12.7 + 7.35i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-29.3 - 50.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (66.1 + 38.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 26.0iT - 1.21e4T^{2} \)
29 \( 1 + (217. + 125. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-190. - 109. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (97.2 - 168. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (37.6 + 65.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-210. + 365. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-191. - 331. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (92.7 - 53.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-266. + 462. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-35.7 + 20.6i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (166. - 288. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 500. iT - 3.57e5T^{2} \)
73 \( 1 + (-351. + 203. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (75.1 + 130. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (493. - 854. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (311. - 539. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (407. + 235. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.50532961441316296456152481362, −11.74823519266409263892961873791, −10.56041786266494045682891876469, −9.570260370297449233273000247849, −8.274667223776260876146864049371, −7.11782822594501397774241753067, −5.80952304514134810377009062455, −4.43791900371274391342823282819, −3.61224442638517162851820467632, −2.21064380442337613989819995052, 0.45664117497516018752812290111, 3.23343068643681783406384601188, 4.08552560948477321309096927612, 5.64024517074106673941608016304, 6.29233879259595012599370448535, 7.46978306085648940184826700621, 8.779689541449201543323775409170, 9.861723932453527412758240024654, 11.01763135153024297929974570636, 12.11323026577306998685074882015

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