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Properties

Label 2-189-63.59-c3-0-4
Degree $2$
Conductor $189$
Sign $0.909 - 0.416i$
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  + (−0.958 − 0.553i)2-s + (−3.38 − 5.86i)4-s − 12.4·5-s + (−18.2 + 2.96i)7-s + 16.3i·8-s + (11.9 + 6.90i)10-s + 3.72i·11-s + (68.0 + 39.2i)13-s + (19.1 + 7.27i)14-s + (−18.0 + 31.2i)16-s + (56.8 − 98.4i)17-s + (33.4 − 19.3i)19-s + (42.2 + 73.1i)20-s + (2.05 − 3.56i)22-s + 37.6i·23-s + ⋯
L(s)  = 1  + (−0.338 − 0.195i)2-s + (−0.423 − 0.733i)4-s − 1.11·5-s + (−0.987 + 0.160i)7-s + 0.722i·8-s + (0.378 + 0.218i)10-s + 0.102i·11-s + (1.45 + 0.838i)13-s + (0.365 + 0.138i)14-s + (−0.281 + 0.488i)16-s + (0.810 − 1.40i)17-s + (0.403 − 0.233i)19-s + (0.472 + 0.818i)20-s + (0.0199 − 0.0345i)22-s + 0.340i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.909 - 0.416i$
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: \((2,\ 189,\ (\ :3/2),\ 0.909 - 0.416i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.724229 + 0.157858i\)
\(L(\frac12)\) \(\approx\) \(0.724229 + 0.157858i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.2 - 2.96i)T \)
good2 \( 1 + (0.958 + 0.553i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 12.4T + 125T^{2} \)
11 \( 1 - 3.72iT - 1.33e3T^{2} \)
13 \( 1 + (-68.0 - 39.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-56.8 + 98.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-33.4 + 19.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 37.6iT - 1.21e4T^{2} \)
29 \( 1 + (-144. + 83.7i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (183. - 106. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-132. - 228. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (190. - 329. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-90.2 - 156. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (77.0 - 133. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-162. - 93.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-234. - 405. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-308. - 178. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-48.3 - 83.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 705. iT - 3.57e5T^{2} \)
73 \( 1 + (631. + 364. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-204. + 355. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (321. + 556. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-74.4 - 128. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (605. - 349. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.82958457103707278173076216021, −11.33731658162109207236404740012, −10.05651157858660560661067471935, −9.267550934765750422282263237624, −8.331943287069984759178661769285, −7.03200571595150950260846975093, −5.86137209373760813030725799395, −4.46779428680418134338455540467, −3.18303456144531790195215486203, −0.985636821859802430769102295072, 0.51137232260174905134784368059, 3.49685166378512342580139009060, 3.82012115180373428486581369357, 5.82692702504934130082482633439, 7.12190994150807379590369026695, 8.099869793546155729636071840895, 8.714058037377475482944448854462, 10.02470933982649352798912912010, 11.01069754197201085901615107104, 12.32131122879248027957466792099

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