Properties

Label 2-189-63.38-c3-0-7
Degree $2$
Conductor $189$
Sign $0.712 + 0.701i$
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  − 5.07i·2-s − 17.7·4-s + (−4.21 + 7.29i)5-s + (−4.29 + 18.0i)7-s + 49.6i·8-s + (37.0 + 21.3i)10-s + (39.7 − 22.9i)11-s + (18.3 − 10.6i)13-s + (91.4 + 21.8i)14-s + 109.·16-s + (8.26 − 14.3i)17-s + (−49.7 + 28.7i)19-s + (74.8 − 129. i)20-s + (−116. − 201. i)22-s + (167. + 96.8i)23-s + ⋯
L(s)  = 1  − 1.79i·2-s − 2.22·4-s + (−0.376 + 0.652i)5-s + (−0.232 + 0.972i)7-s + 2.19i·8-s + (1.17 + 0.676i)10-s + (1.08 − 0.628i)11-s + (0.392 − 0.226i)13-s + (1.74 + 0.416i)14-s + 1.71·16-s + (0.117 − 0.204i)17-s + (−0.600 + 0.346i)19-s + (0.837 − 1.45i)20-s + (−1.12 − 1.95i)22-s + (1.52 + 0.877i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.712 + 0.701i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.712 + 0.701i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.17126 - 0.480092i\)
\(L(\frac12)\) \(\approx\) \(1.17126 - 0.480092i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.29 - 18.0i)T \)
good2 \( 1 + 5.07iT - 8T^{2} \)
5 \( 1 + (4.21 - 7.29i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-39.7 + 22.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-18.3 + 10.6i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-8.26 + 14.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (49.7 - 28.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-167. - 96.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-47.1 - 27.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 294. iT - 2.97e4T^{2} \)
37 \( 1 + (-185. - 320. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (166. + 287. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 419.T + 1.03e5T^{2} \)
53 \( 1 + (-275. - 159. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 298.T + 2.05e5T^{2} \)
61 \( 1 - 226. iT - 2.26e5T^{2} \)
67 \( 1 + 99.7T + 3.00e5T^{2} \)
71 \( 1 + 176. iT - 3.57e5T^{2} \)
73 \( 1 + (-142. - 82.2i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 374.T + 4.93e5T^{2} \)
83 \( 1 + (457. - 792. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (67.8 + 117. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (377. + 217. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−11.76373535533439227702286758930, −11.21105800804125873151920168674, −10.30160039162990749317800669199, −9.145663376776289312836791848161, −8.559727024358292803302811425552, −6.70412470732800003832907184938, −5.19452453450264095681996590171, −3.61603829924361167674474358163, −2.93138458625007207555352374818, −1.30450405917732960993218469331, 0.67432804403257040026858539788, 4.09371015643040528859721392785, 4.68548795171869343779307752435, 6.27680190588581735976023193630, 6.95162669772234694329904526551, 7.985914440412572480115889972276, 8.894252291596093220815058767446, 9.752954166264982513484109156865, 11.25615434187081442443155411867, 12.75927531912731930049991514242

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