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Properties

Label 2-189-63.4-c3-0-19
Degree $2$
Conductor $189$
Sign $-0.464 + 0.885i$
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  + (2.10 − 3.65i)2-s + (−4.88 − 8.45i)4-s + 7.82·5-s + (16.1 − 9.02i)7-s − 7.43·8-s + (16.4 − 28.5i)10-s + 12.8·11-s + (6.74 − 11.6i)13-s + (1.15 − 78.0i)14-s + (23.3 − 40.5i)16-s + (−35.5 + 61.4i)17-s + (−46.2 − 80.0i)19-s + (−38.1 − 66.1i)20-s + (27.0 − 46.8i)22-s − 3.94·23-s + ⋯
L(s)  = 1  + (0.745 − 1.29i)2-s + (−0.610 − 1.05i)4-s + 0.699·5-s + (0.873 − 0.487i)7-s − 0.328·8-s + (0.521 − 0.903i)10-s + 0.352·11-s + (0.143 − 0.249i)13-s + (0.0220 − 1.48i)14-s + (0.365 − 0.632i)16-s + (−0.506 + 0.877i)17-s + (−0.558 − 0.966i)19-s + (−0.427 − 0.739i)20-s + (0.262 − 0.454i)22-s − 0.0357·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.464 + 0.885i$
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: \((2,\ 189,\ (\ :3/2),\ -0.464 + 0.885i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.60444 - 2.65265i\)
\(L(\frac12)\) \(\approx\) \(1.60444 - 2.65265i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.1 + 9.02i)T \)
good2 \( 1 + (-2.10 + 3.65i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 7.82T + 125T^{2} \)
11 \( 1 - 12.8T + 1.33e3T^{2} \)
13 \( 1 + (-6.74 + 11.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (35.5 - 61.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (46.2 + 80.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 3.94T + 1.21e4T^{2} \)
29 \( 1 + (-90.3 - 156. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (135. + 234. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-110. - 190. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-33.6 + 58.2i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-237. - 411. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (256. - 444. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-238. + 413. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-358. - 621. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (188. - 326. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (347. + 601. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 230.T + 3.57e5T^{2} \)
73 \( 1 + (258. - 446. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-471. + 817. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-84.4 - 146. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-149. - 258. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-389. - 674. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.65577232739242892173050149839, −10.98072891445212125271577639297, −10.23442064473292494230864743221, −9.097263346249864643058042358579, −7.74775987414477456991434687491, −6.21290347115818278376180537697, −4.89039951752557873731061893952, −3.97281353135393631904063701682, −2.42122921699324557200800585981, −1.28276784084793684700893861779, 1.96146737324392715890022560228, 4.08431443614943470986450649442, 5.21977202210348135963780374443, 6.01234984012388028573712646603, 7.04980868995903591539043698766, 8.162174878741868584880831898747, 9.104802887975361913420811426135, 10.45253467411497657688531159967, 11.67517347536661970684554441341, 12.71346177362995988102021735309

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