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Properties

Label 2-189-7.2-c3-0-9
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: $\chi_{189} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.550 - 0.834i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.357791 + 0.664884i\)
\(L(\frac12)\) \(\approx\) \(0.357791 + 0.664884i\)
\(L(\frac{5}{2})\) 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

−12.87735576177266559093735021616, −10.89680082827782186922795659187, −10.15636881542475604241666389550, −9.207382601688159548893463223858, −8.223084439362166806767897501558, −7.42877386335435633841461829768, −6.22784683997926844683193028348, −5.67204312652271760719729679775, −3.90311505108765299014561907436, −0.965780004949306879514919697094, 0.64346038684162800878137676299, 2.39497413696721882004309642221, 3.34217234789234225204990978917, 4.83679382966102535207946364842, 6.87905317277454545791515780409, 8.218660157973910262449816696402, 9.054856604802664092198099525786, 9.969001163177901794789942188506, 10.57075777011696155496618988141, 11.84242167262619989168104412853

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