↑

Properties

Label 2-171-19.8-c2-0-6
Degree $2$
Conductor $171$
Sign $-0.307 - 0.951i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.90 + 1.67i)2-s + (3.63 + 6.29i)4-s + (−3.47 + 6.01i)5-s − 1.22·7-s + 10.9i·8-s + (−20.1 + 11.6i)10-s + 0.0363·11-s + (14.6 − 8.44i)13-s + (−3.57 − 2.06i)14-s + (−3.86 + 6.68i)16-s + (4.59 − 7.95i)17-s + (12.7 + 14.0i)19-s − 50.4·20-s + (0.105 + 0.0610i)22-s + (4.87 + 8.45i)23-s + ⋯
L(s)  = 1  + (1.45 + 0.839i)2-s + (0.908 + 1.57i)4-s + (−0.694 + 1.20i)5-s − 0.175·7-s + 1.36i·8-s + (−2.01 + 1.16i)10-s + 0.00330·11-s + (1.12 − 0.649i)13-s + (−0.255 − 0.147i)14-s + (−0.241 + 0.418i)16-s + (0.270 − 0.467i)17-s + (0.672 + 0.739i)19-s − 2.52·20-s + (0.00480 + 0.00277i)22-s + (0.212 + 0.367i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.307 - 0.951i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.307 - 0.951i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.68327 + 2.31282i\)
\(L(\frac12)\) \(\approx\) \(1.68327 + 2.31282i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-12.7 - 14.0i)T \)
good2 \( 1 + (-2.90 - 1.67i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (3.47 - 6.01i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 1.22T + 49T^{2} \)
11 \( 1 - 0.0363T + 121T^{2} \)
13 \( 1 + (-14.6 + 8.44i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-4.59 + 7.95i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-4.87 - 8.45i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (4.50 - 2.60i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 44.3iT - 961T^{2} \)
37 \( 1 - 45.5iT - 1.36e3T^{2} \)
41 \( 1 + (50.0 + 28.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-15.1 + 26.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.5 + 44.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (10.0 - 5.82i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-17.7 - 10.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.23 + 14.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1.83 - 1.05i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-33.7 - 19.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (28.1 - 48.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-39.8 - 23.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 65.3T + 6.88e3T^{2} \)
89 \( 1 + (-134. + 77.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (98.8 + 57.0i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−13.16090083242179031579155337603, −11.95090782184439486094948794673, −11.25916523400845975193720238393, −10.02654255364769892736362586666, −8.145492683398134517619013325676, −7.29147826710482008208108365821, −6.38532885632746590634840985963, −5.37780107723250686101730354464, −3.79581353239391877475427114118, −3.13533688015254587871084100114, 1.35828423939197113323052220288, 3.31461941382714965561897227577, 4.35101180684596319595704957287, 5.22891759327037590617749421412, 6.51314973361385856704051875596, 8.214273428476170194821913199479, 9.269020513608400689430012499315, 10.76289134422589092821422561582, 11.55868608046566378249252728268, 12.36906541395567595813378453464

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