↑

Properties

Label 2-189-63.5-c3-0-15
Degree $2$
Conductor $189$
Sign $-0.252 + 0.967i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.91i·2-s − 0.495·4-s + (−8.60 − 14.9i)5-s + (−4.20 + 18.0i)7-s + 21.8i·8-s + (43.4 − 25.0i)10-s + (−30.9 − 17.8i)11-s + (8.99 + 5.19i)13-s + (−52.5 − 12.2i)14-s − 67.7·16-s + (−64.1 − 111. i)17-s + (−84.7 − 48.9i)19-s + (4.26 + 7.38i)20-s + (52.0 − 90.0i)22-s + (−22.1 + 12.7i)23-s + ⋯
L(s)  = 1  + 1.03i·2-s − 0.0619·4-s + (−0.769 − 1.33i)5-s + (−0.227 + 0.973i)7-s + 0.966i·8-s + (1.37 − 0.793i)10-s + (−0.847 − 0.489i)11-s + (0.191 + 0.110i)13-s + (−1.00 − 0.234i)14-s − 1.05·16-s + (−0.915 − 1.58i)17-s + (−1.02 − 0.590i)19-s + (0.0476 + 0.0825i)20-s + (0.504 − 0.873i)22-s + (−0.200 + 0.115i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.161764 - 0.209490i\)
\(L(\frac12)\) \(\approx\) \(0.161764 - 0.209490i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.20 - 18.0i)T \)
good2 \( 1 - 2.91iT - 8T^{2} \)
5 \( 1 + (8.60 + 14.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (30.9 + 17.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-8.99 - 5.19i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (64.1 + 111. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (84.7 + 48.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (22.1 - 12.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-14.1 + 8.16i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 185. iT - 2.97e4T^{2} \)
37 \( 1 + (0.462 - 0.801i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (231. - 401. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (43.2 + 74.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 287.T + 1.03e5T^{2} \)
53 \( 1 + (334. - 193. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 90.4T + 2.05e5T^{2} \)
61 \( 1 - 139. iT - 2.26e5T^{2} \)
67 \( 1 - 303.T + 3.00e5T^{2} \)
71 \( 1 - 351. iT - 3.57e5T^{2} \)
73 \( 1 + (-761. + 439. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 14.2T + 4.93e5T^{2} \)
83 \( 1 + (125. + 217. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (505. - 875. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.12e3 - 649. i)T + (4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.81357408456405462886773206195, −11.16495145953107666959184694485, −9.301780025125396918473249752905, −8.553893207816570903838963036084, −7.86315363167273866806136267731, −6.57661484918128912190246481060, −5.41091374303155843330544706956, −4.61627455193211776275682020624, −2.52660650548357319871967839304, −0.10669011777848808852239708978, 2.04255102076624162636456388341, 3.39384867080722723137323790391, 4.16012359466569325218102501555, 6.44914614967348934180767961403, 7.14697147812365566613385058003, 8.283885584237780983337810045121, 10.11707181884817206570030759484, 10.64558197377749659811573056662, 11.02610000651233056655471320919, 12.33781407089634730816621269387

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