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Properties

Label 2-189-63.4-c3-0-2
Degree $2$
Conductor $189$
Sign $-0.816 + 0.577i$
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  + (−1.68 + 2.92i)2-s + (−1.68 − 2.92i)4-s + 9.74·5-s + (0.158 − 18.5i)7-s − 15.5·8-s + (−16.4 + 28.4i)10-s − 38.9·11-s + (−31.2 + 54.1i)13-s + (53.8 + 31.6i)14-s + (39.8 − 68.9i)16-s + (−63.3 + 109. i)17-s + (22.7 + 39.3i)19-s + (−16.4 − 28.5i)20-s + (65.6 − 113. i)22-s − 153.·23-s + ⋯
L(s)  = 1  + (−0.596 + 1.03i)2-s + (−0.211 − 0.365i)4-s + 0.871·5-s + (0.00854 − 0.999i)7-s − 0.688·8-s + (−0.519 + 0.900i)10-s − 1.06·11-s + (−0.667 + 1.15i)13-s + (1.02 + 0.605i)14-s + (0.621 − 1.07i)16-s + (−0.903 + 1.56i)17-s + (0.274 + 0.475i)19-s + (−0.184 − 0.318i)20-s + (0.636 − 1.10i)22-s − 1.39·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.816 + 0.577i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.136427 - 0.428702i\)
\(L(\frac12)\) \(\approx\) \(0.136427 - 0.428702i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.158 + 18.5i)T \)
good2 \( 1 + (1.68 - 2.92i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 9.74T + 125T^{2} \)
11 \( 1 + 38.9T + 1.33e3T^{2} \)
13 \( 1 + (31.2 - 54.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (63.3 - 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-22.7 - 39.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 153.T + 1.21e4T^{2} \)
29 \( 1 + (27.4 + 47.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (54.8 + 95.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-144. - 250. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-11.3 + 19.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-23.9 - 41.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-193. + 334. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-133. + 231. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-193. - 334. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (17.8 - 30.9i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (17.8 + 30.9i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 146.T + 3.57e5T^{2} \)
73 \( 1 + (364. - 631. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (250. - 434. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (169. + 293. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-104. - 180. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (56.7 + 98.2i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.94329268727761161613818705804, −11.65753195687125272348756919416, −10.28729910676759628134244552784, −9.717749917147972239698165869952, −8.445694778662083091062753248101, −7.59840625149401740577953309012, −6.58123023260123238789645619946, −5.71477782027553849614654720576, −4.12013276004571167327305572178, −2.09605508814524559816946870011, 0.21621418732656702038164563775, 2.21076535128012499339832729415, 2.80661983807514210074634800659, 5.16768084538084428825903600889, 5.97527145474657189453786897049, 7.65383171937278806525843469923, 8.956686084344554169399148972250, 9.644660317892015383146194999587, 10.43750461615427924942225394819, 11.39873224896114066241850827041

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