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Properties

Label 2-171-171.50-c1-0-8
Degree $2$
Conductor $171$
Sign $0.923 - 0.382i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.345 + 0.598i)2-s + (0.763 − 1.55i)3-s + (0.760 + 1.31i)4-s − 0.106i·5-s + (0.666 + 0.994i)6-s + (1.77 + 3.08i)7-s − 2.43·8-s + (−1.83 − 2.37i)9-s + (0.0640 + 0.0369i)10-s + (3.79 − 2.19i)11-s + (2.63 − 0.176i)12-s + (−0.360 + 0.208i)13-s − 2.46·14-s + (−0.166 − 0.0816i)15-s + (−0.680 + 1.17i)16-s + (3.21 − 1.85i)17-s + ⋯
L(s)  = 1  + (−0.244 + 0.423i)2-s + (0.440 − 0.897i)3-s + (0.380 + 0.659i)4-s − 0.0478i·5-s + (0.272 + 0.406i)6-s + (0.672 + 1.16i)7-s − 0.860·8-s + (−0.611 − 0.791i)9-s + (0.0202 + 0.0116i)10-s + (1.14 − 0.661i)11-s + (0.759 − 0.0509i)12-s + (−0.100 + 0.0577i)13-s − 0.657·14-s + (−0.0429 − 0.0210i)15-s + (−0.170 + 0.294i)16-s + (0.780 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26985 + 0.252697i\)
\(L(\frac12)\) \(\approx\) \(1.26985 + 0.252697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.763 + 1.55i)T \)
19 \( 1 + (4.10 + 1.45i)T \)
good2 \( 1 + (0.345 - 0.598i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 0.106iT - 5T^{2} \)
7 \( 1 + (-1.77 - 3.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.79 + 2.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.360 - 0.208i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.21 + 1.85i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.68 - 2.70i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.50T + 29T^{2} \)
31 \( 1 + (3.01 + 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.84iT - 37T^{2} \)
41 \( 1 + 9.97T + 41T^{2} \)
43 \( 1 + (-5.40 + 9.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.251iT - 47T^{2} \)
53 \( 1 + (1.44 - 2.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.37T + 59T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 + (-9.17 + 5.29i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.43 - 9.41i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.47 - 4.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.947 - 0.547i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.21 + 4.16i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.61 - 4.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.3 - 6.55i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.58775584781877494146578589987, −11.96783104016684004407676531026, −11.29404619482811720392051612202, −9.174399370486480053792966515637, −8.638476619086272083026571111265, −7.73842050531660332734585066279, −6.65844933187353452211110791685, −5.68316759477406983949322403333, −3.49356482529168087488184159699, −2.07374228681595344485725021708, 1.76461641896743622825302582077, 3.67767855385367395179589967314, 4.77110205409693091970121227636, 6.30589914694600220969178824499, 7.65599420231400282945366318040, 8.920646577345967624070988243432, 9.943067819539208713347020126219, 10.56737035666491457398740936439, 11.31696440278383705216603264996, 12.50954612627464105757638718908

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