Properties

Label 2-171-19.12-c2-0-12
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: $\chi_{171} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.0752 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.38475 - 2.21167i\)
\(L(\frac12)\) \(\approx\) \(2.38475 - 2.21167i\)
\(L(2)\) 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

−12.17664227569489443543533581589, −11.83362275136858106661015131784, −10.53132334123534203489148791913, −9.756114051529023828084166096806, −8.096105771523402113012550864843, −6.52420465599889697651304507444, −5.47694688333101171389857247221, −4.33581256701005924747688623182, −3.28428691420936871192945215184, −1.67087352388340331862219089436, 2.73491459731614340969944701637, 4.19481637671908257042329553979, 5.02183892025749073444846638835, 6.48567954033792575814636304195, 7.00105050116020131892372748738, 8.295374986613224253739944189770, 9.683433439404635297119233399492, 11.30604362630941493385148390855, 12.09169331568405685298591842387, 12.85779396983088886822758878485

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