Properties

Label 2-189-9.4-c3-0-11
Degree $2$
Conductor $189$
Sign $0.0376 - 0.999i$
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  + (2.28 + 3.95i)2-s + (−6.41 + 11.1i)4-s + (10.3 − 18.0i)5-s + (3.5 + 6.06i)7-s − 22.0·8-s + 94.8·10-s + (22.6 + 39.3i)11-s + (−14.6 + 25.3i)13-s + (−15.9 + 27.6i)14-s + (1.04 + 1.81i)16-s + 98.0·17-s + 31.1·19-s + (133. + 230. i)20-s + (−103. + 179. i)22-s + (4.19 − 7.26i)23-s + ⋯
L(s)  = 1  + (0.806 + 1.39i)2-s + (−0.801 + 1.38i)4-s + (0.929 − 1.61i)5-s + (0.188 + 0.327i)7-s − 0.973·8-s + 3.00·10-s + (0.622 + 1.07i)11-s + (−0.312 + 0.541i)13-s + (−0.304 + 0.528i)14-s + (0.0163 + 0.0283i)16-s + 1.39·17-s + 0.376·19-s + (1.49 + 2.58i)20-s + (−1.00 + 1.73i)22-s + (0.0380 − 0.0658i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0376 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.0376 - 0.999i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.0376 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.27145 + 2.18758i\)
\(L(\frac12)\) \(\approx\) \(2.27145 + 2.18758i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (-2.28 - 3.95i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-10.3 + 18.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-22.6 - 39.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (14.6 - 25.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 98.0T + 4.91e3T^{2} \)
19 \( 1 - 31.1T + 6.85e3T^{2} \)
23 \( 1 + (-4.19 + 7.26i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (36.1 + 62.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-15.4 + 26.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 196.T + 5.06e4T^{2} \)
41 \( 1 + (-106. + 184. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (118. + 205. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (110. + 190. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 55.9T + 1.48e5T^{2} \)
59 \( 1 + (327. - 566. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (174. + 302. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (105. - 182. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 548.T + 3.57e5T^{2} \)
73 \( 1 + 266.T + 3.89e5T^{2} \)
79 \( 1 + (-134. - 233. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-312. - 541. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + (-145. - 252. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−12.44577645507283483112273449728, −12.14270865570122696783036113181, −9.926060750794621875328735967812, −9.126597625282542699107332778248, −8.136062604234353836457832335779, −7.00840082774838456010566787908, −5.75645136214512704186470172380, −5.11873000888560978627915622638, −4.18279758742032238357193203156, −1.62853322429164973360612516618, 1.40803236426816682296940331785, 2.92755198575027433565321772892, 3.49375895026817068323649150992, 5.30361256739707124590441537258, 6.28956904272525972407708421297, 7.64095142443614679199664995921, 9.532089086520855403475989408724, 10.27015919862094400827925379976, 10.96063107839544459993322784055, 11.67608273299375011566864352921

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