↑

Properties

Label 2-171-171.121-c1-0-2
Degree $2$
Conductor $171$
Sign $-0.913 - 0.407i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.888 + 1.53i)2-s + (1.15 + 1.29i)3-s + (−0.580 − 1.00i)4-s − 1.27·5-s + (−3.01 + 0.632i)6-s + (0.657 + 1.13i)7-s − 1.49·8-s + (−0.328 + 2.98i)9-s + (1.13 − 1.97i)10-s + (−0.130 − 0.225i)11-s + (0.626 − 1.91i)12-s + (0.933 + 1.61i)13-s − 2.33·14-s + (−1.47 − 1.65i)15-s + (2.48 − 4.30i)16-s + (−0.0508 − 0.0880i)17-s + ⋯
L(s)  = 1  + (−0.628 + 1.08i)2-s + (0.667 + 0.744i)3-s + (−0.290 − 0.502i)4-s − 0.572·5-s + (−1.23 + 0.258i)6-s + (0.248 + 0.430i)7-s − 0.527·8-s + (−0.109 + 0.993i)9-s + (0.359 − 0.623i)10-s + (−0.0392 − 0.0679i)11-s + (0.180 − 0.551i)12-s + (0.259 + 0.448i)13-s − 0.625·14-s + (−0.381 − 0.426i)15-s + (0.621 − 1.07i)16-s + (−0.0123 − 0.0213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.913 - 0.407i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192313 + 0.903999i\)
\(L(\frac12)\) \(\approx\) \(0.192313 + 0.903999i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.15 - 1.29i)T \)
19 \( 1 + (-3.11 + 3.05i)T \)
good2 \( 1 + (0.888 - 1.53i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.27T + 5T^{2} \)
7 \( 1 + (-0.657 - 1.13i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.130 + 0.225i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.933 - 1.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0508 + 0.0880i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.611 - 1.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.52T + 29T^{2} \)
31 \( 1 + (0.617 - 1.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.59T + 37T^{2} \)
41 \( 1 - 8.21T + 41T^{2} \)
43 \( 1 + (1.53 - 2.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 + (2.59 - 4.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.02T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + (-0.390 - 0.677i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.19 + 14.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.397 + 0.687i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.82 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.03 + 5.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.75 + 9.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.83 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−13.55635143212553524067033835534, −12.04322836274160464673318763292, −11.11099287707873104318213939993, −9.701368725438947526385618966310, −8.943390978497950387488993093866, −8.107350303226892132203434698651, −7.29855179645679625552067152897, −5.86534893821224232023129350868, −4.49980472995665387020075894882, −2.95526978905629866578017899439, 1.07977128382653303513199561404, 2.68999897758983145279848865285, 3.91245971269105548870939262201, 6.07684047458979637102966288468, 7.54716545712844921365674866571, 8.285468355230217303002537146877, 9.376511642780686241522517726603, 10.34988407143960343153285619410, 11.42426070914637046203425234397, 12.15081228976354258503329819212

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