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Properties

Label 2-189-63.20-c3-0-2
Degree $2$
Conductor $189$
Sign $0.789 - 0.613i$
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  + (−4.27 − 2.46i)2-s + (8.16 + 14.1i)4-s + (7.65 + 13.2i)5-s + (6.32 − 17.4i)7-s − 41.0i·8-s − 75.5i·10-s + (36.9 + 21.3i)11-s + (−24.2 + 14.0i)13-s + (−69.9 + 58.7i)14-s + (−35.9 + 62.2i)16-s − 82.0·17-s + 113. i·19-s + (−125. + 216. i)20-s + (−105. − 182. i)22-s + (25.2 − 14.5i)23-s + ⋯
L(s)  = 1  + (−1.51 − 0.871i)2-s + (1.02 + 1.76i)4-s + (0.684 + 1.18i)5-s + (0.341 − 0.939i)7-s − 1.81i·8-s − 2.38i·10-s + (1.01 + 0.584i)11-s + (−0.517 + 0.299i)13-s + (−1.33 + 1.12i)14-s + (−0.562 + 0.973i)16-s − 1.17·17-s + 1.36i·19-s + (−1.39 + 2.42i)20-s + (−1.01 − 1.76i)22-s + (0.228 − 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.789 - 0.613i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.789 - 0.613i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.789741 + 0.270897i\)
\(L(\frac12)\) \(\approx\) \(0.789741 + 0.270897i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.32 + 17.4i)T \)
good2 \( 1 + (4.27 + 2.46i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-7.65 - 13.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-36.9 - 21.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (24.2 - 14.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 82.0T + 4.91e3T^{2} \)
19 \( 1 - 113. iT - 6.85e3T^{2} \)
23 \( 1 + (-25.2 + 14.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (16.5 + 9.56i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-97.0 + 56.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 69.2T + 5.06e4T^{2} \)
41 \( 1 + (-242. - 419. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (194. - 337. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-53.4 + 92.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 207. iT - 1.48e5T^{2} \)
59 \( 1 + (-97.8 - 169. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-446. - 257. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-221. - 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 640. iT - 3.57e5T^{2} \)
73 \( 1 - 528. iT - 3.89e5T^{2} \)
79 \( 1 + (-20.7 + 35.8i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (554. - 960. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 121.T + 7.04e5T^{2} \)
97 \( 1 + (748. + 432. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.61067118976384507230523892011, −11.07046085923179752688711557675, −9.981846865435069903167883961531, −9.746963585972337653715924248976, −8.344087155594092642816755750196, −7.22839525852954326248130461183, −6.50951333783169205134370122195, −4.10063547229906116993271554684, −2.56996781241155470397638534895, −1.42679335814904990960303104137, 0.64513771492367484492715866718, 2.04232355955255057886558625820, 4.91605113887828260009132529977, 5.91127017542577253451690468197, 6.93438407722834366310187631610, 8.331085199672194010499949314245, 9.123137318973382522401136711256, 9.247824336999200285740642943622, 10.73669184070102774048249619998, 11.79241618095413595011135414140

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