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Properties

Label 2-189-63.59-c3-0-6
Degree $2$
Conductor $189$
Sign $0.394 + 0.919i$
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  + (−3.22 − 1.86i)2-s + (2.92 + 5.06i)4-s − 13.6·5-s + (−10.0 + 15.5i)7-s + 8.02i·8-s + (44.0 + 25.4i)10-s − 24.6i·11-s + (−2.03 − 1.17i)13-s + (61.4 − 31.2i)14-s + (38.3 − 66.3i)16-s + (−63.8 + 110. i)17-s + (87.2 − 50.3i)19-s + (−39.9 − 69.1i)20-s + (−45.7 + 79.2i)22-s − 48.5i·23-s + ⋯
L(s)  = 1  + (−1.13 − 0.657i)2-s + (0.365 + 0.632i)4-s − 1.22·5-s + (−0.545 + 0.838i)7-s + 0.354i·8-s + (1.39 + 0.803i)10-s − 0.674i·11-s + (−0.0434 − 0.0250i)13-s + (1.17 − 0.596i)14-s + (0.598 − 1.03i)16-s + (−0.910 + 1.57i)17-s + (1.05 − 0.608i)19-s + (−0.446 − 0.773i)20-s + (−0.443 + 0.768i)22-s − 0.439i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.394 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.394 + 0.919i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.394 + 0.919i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.405287 - 0.267163i\)
\(L(\frac12)\) \(\approx\) \(0.405287 - 0.267163i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.0 - 15.5i)T \)
good2 \( 1 + (3.22 + 1.86i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 13.6T + 125T^{2} \)
11 \( 1 + 24.6iT - 1.33e3T^{2} \)
13 \( 1 + (2.03 + 1.17i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (63.8 - 110. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-87.2 + 50.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 48.5iT - 1.21e4T^{2} \)
29 \( 1 + (-171. + 99.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (37.2 - 21.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (42.0 + 72.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-92.8 + 160. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (185. + 321. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-252. + 437. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-372. - 215. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-156. - 270. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-475. - 274. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-325. - 564. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 18.9iT - 3.57e5T^{2} \)
73 \( 1 + (-525. - 303. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (217. - 377. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-424. - 734. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (147. + 254. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (900. - 520. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.72673621556263496085890030641, −10.92809139584798512551909839140, −9.981622340914754830635145465540, −8.637281181425574057648306551555, −8.486181968685249894542717444686, −7.05829827181211237399396032900, −5.57176337529064925749366898851, −3.82420234066055828103751377231, −2.46006439680522042524496805369, −0.52101011890733760366662820250, 0.74962340687123081374281688659, 3.40522089078886165603087135742, 4.64867392598204206046100338870, 6.65789144931174852241756108828, 7.35955281204981508810535561029, 8.006128374390838821739657542196, 9.287061099972862536409468564205, 9.966640250271574440739906857586, 11.16661541117667473694625394530, 12.13403387769502580437947736155

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