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Properties

Label 2-189-7.4-c3-0-5
Degree $2$
Conductor $189$
Sign $-0.550 + 0.834i$
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  + (2.73 + 4.73i)2-s + (−10.9 + 18.9i)4-s + (0.0995 + 0.172i)5-s + (−16.0 + 9.31i)7-s − 75.5·8-s + (−0.543 + 0.941i)10-s + (14.2 − 24.6i)11-s + 32.5·13-s + (−87.7 − 50.2i)14-s + (−118. − 206. i)16-s + (−57.7 + 100. i)17-s + (−10.5 − 18.3i)19-s − 4.34·20-s + 155.·22-s + (46.8 + 81.2i)23-s + ⋯
L(s)  = 1  + (0.965 + 1.67i)2-s + (−1.36 + 2.36i)4-s + (0.00890 + 0.0154i)5-s + (−0.864 + 0.503i)7-s − 3.33·8-s + (−0.0171 + 0.0297i)10-s + (0.389 − 0.674i)11-s + 0.695·13-s + (−1.67 − 0.959i)14-s + (−1.85 − 3.21i)16-s + (−0.823 + 1.42i)17-s + (−0.127 − 0.221i)19-s − 0.0486·20-s + 1.50·22-s + (0.425 + 0.736i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.550 + 0.834i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.550 + 0.834i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.801352 - 1.48915i\)
\(L(\frac12)\) \(\approx\) \(0.801352 - 1.48915i\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.0 - 9.31i)T \)
good2 \( 1 + (-2.73 - 4.73i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-0.0995 - 0.172i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-14.2 + 24.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 32.5T + 2.19e3T^{2} \)
17 \( 1 + (57.7 - 100. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (10.5 + 18.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-46.8 - 81.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 231.T + 2.43e4T^{2} \)
31 \( 1 + (140. - 243. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-73.2 - 126. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 111.T + 6.89e4T^{2} \)
43 \( 1 - 392.T + 7.95e4T^{2} \)
47 \( 1 + (-136. - 236. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (170. - 294. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (348. - 603. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-185. - 320. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (43.5 - 75.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 88.3T + 3.57e5T^{2} \)
73 \( 1 + (-401. + 695. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (182. + 315. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 921.T + 5.71e5T^{2} \)
89 \( 1 + (105. + 183. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 845.T + 9.12e5T^{2} \)
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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

−12.99845167477971908771356115500, −12.38937996825872917402450390221, −10.98097827261234000268232195051, −9.090915533305979560240234662751, −8.629163887843692456864901078547, −7.31376910940433603597690491136, −6.24497476392337046211159272760, −5.77382350189906130637586280122, −4.23528249259268308258170877289, −3.22755874584627103530527762770, 0.54102078373035419820936887699, 2.20020838318055760737312767104, 3.52217803080998295577186959819, 4.42673692821513344861657515633, 5.72086715775267218234658736742, 6.99595800971150838431285950785, 9.225184162316953500330907099064, 9.594739068051937644338758278284, 10.91214088761254993040081533127, 11.33985291212287500990024800494

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