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Properties

Label 2-171-19.8-c2-0-2
Degree $2$
Conductor $171$
Sign $0.0752 - 0.997i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.05 − 1.76i)2-s + (4.23 + 7.34i)4-s + (0.533 − 0.923i)5-s + 0.477·7-s − 15.8i·8-s + (−3.26 + 1.88i)10-s − 11.1·11-s + (−12.9 + 7.49i)13-s + (−1.45 − 0.842i)14-s + (−10.9 + 19.0i)16-s + (−6.11 + 10.5i)17-s + (−6.52 − 17.8i)19-s + 9.03·20-s + (34.1 + 19.7i)22-s + (20.7 + 35.9i)23-s + ⋯
L(s)  = 1  + (−1.52 − 0.883i)2-s + (1.05 + 1.83i)4-s + (0.106 − 0.184i)5-s + 0.0681·7-s − 1.97i·8-s + (−0.326 + 0.188i)10-s − 1.01·11-s + (−0.998 + 0.576i)13-s + (−0.104 − 0.0602i)14-s + (−0.686 + 1.18i)16-s + (−0.359 + 0.623i)17-s + (−0.343 − 0.939i)19-s + 0.451·20-s + (1.55 + 0.896i)22-s + (0.901 + 1.56i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.0752 - 0.997i$
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: \((2,\ 171,\ (\ :1),\ 0.0752 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.187344 + 0.173747i\)
\(L(\frac12)\) \(\approx\) \(0.187344 + 0.173747i\)
\(L(2)\) 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 \( 1 + (6.52 + 17.8i)T \)
good2 \( 1 + (3.05 + 1.76i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-0.533 + 0.923i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 0.477T + 49T^{2} \)
11 \( 1 + 11.1T + 121T^{2} \)
13 \( 1 + (12.9 - 7.49i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (6.11 - 10.5i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-20.7 - 35.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (30.5 - 17.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 15.8iT - 961T^{2} \)
37 \( 1 + 24.3iT - 1.36e3T^{2} \)
41 \( 1 + (5.83 + 3.37i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (31.6 - 54.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-20.2 - 35.0i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-26.3 + 15.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (69.1 + 39.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (35.8 + 62.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-7.62 + 4.40i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (42.2 + 24.4i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (56.7 - 98.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-104. - 60.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 65.4T + 6.88e3T^{2} \)
89 \( 1 + (-133. + 76.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (83.0 + 47.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.55007602241397457517311054511, −11.29823185845432265725190970007, −10.81821734857478988243950292295, −9.592697195960500559416243897395, −9.050357660626387824996333817721, −7.85057839422736620500057547756, −7.02655907803074792130908284272, −5.05336319159583949890029035533, −3.09350933367236979500708433694, −1.73756186639596990540366529544, 0.24189237291714546427302848096, 2.42019570254218475076922660464, 5.02220131172903057413462541666, 6.26356820134833897891277118062, 7.34036583006151504487141586315, 8.110257755117325961199771852059, 9.097256456941530181370382504519, 10.25341369092708825585338293772, 10.61406740514983785527602276935, 12.12072590194717112581427354718

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