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Properties

Label 2-171-171.50-c1-0-5
Degree $2$
Conductor $171$
Sign $0.738 - 0.674i$
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  + (−1.07 + 1.86i)2-s + (−1.60 − 0.651i)3-s + (−1.31 − 2.27i)4-s − 3.02i·5-s + (2.93 − 2.28i)6-s + (1.86 + 3.23i)7-s + 1.34·8-s + (2.15 + 2.09i)9-s + (5.63 + 3.25i)10-s + (4.41 − 2.54i)11-s + (0.625 + 4.50i)12-s + (1.79 − 1.03i)13-s − 8.04·14-s + (−1.97 + 4.85i)15-s + (1.17 − 2.04i)16-s + (−4.50 + 2.60i)17-s + ⋯
L(s)  = 1  + (−0.760 + 1.31i)2-s + (−0.926 − 0.376i)3-s + (−0.656 − 1.13i)4-s − 1.35i·5-s + (1.19 − 0.934i)6-s + (0.706 + 1.22i)7-s + 0.475·8-s + (0.717 + 0.696i)9-s + (1.78 + 1.02i)10-s + (1.33 − 0.768i)11-s + (0.180 + 1.30i)12-s + (0.497 − 0.287i)13-s − 2.14·14-s + (−0.508 + 1.25i)15-s + (0.294 − 0.510i)16-s + (−1.09 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.738 - 0.674i$
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: \((2,\ 171,\ (\ :1/2),\ 0.738 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613317 + 0.237775i\)
\(L(\frac12)\) \(\approx\) \(0.613317 + 0.237775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.651i)T \)
19 \( 1 + (-4.31 - 0.593i)T \)
good2 \( 1 + (1.07 - 1.86i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.02iT - 5T^{2} \)
7 \( 1 + (-1.86 - 3.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.41 + 2.54i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.79 + 1.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.50 - 2.60i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.49 + 3.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.61T + 29T^{2} \)
31 \( 1 + (1.31 + 0.758i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.54iT - 37T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + (-0.973 + 1.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.42iT - 47T^{2} \)
53 \( 1 + (4.49 - 7.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.20T + 59T^{2} \)
61 \( 1 + 7.75T + 61T^{2} \)
67 \( 1 + (0.569 - 0.329i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.68 + 4.64i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.90 + 3.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.08 - 5.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.06 - 2.34i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.49 + 2.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.72 + 2.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.70530230762276875181297643306, −11.95455915477409719718057217671, −11.00547630996291952161428183858, −9.073417347023082729231203809832, −8.869889660580356636720669660881, −7.81139032616973651531613780726, −6.38931447623020781325531252444, −5.70198234160685647997607532456, −4.74347530399451456596851706302, −1.16073284949387209206394199402, 1.36810975566757809869277642157, 3.39358548797058453125497635085, 4.50633605041195913328134770085, 6.58966293365102992805864907485, 7.28884027381959891972693086741, 9.187379443392955624240670839575, 9.959825173286257287533769825463, 10.95153363743053965709678859054, 11.21966522004933803580653943175, 11.95325991105388539089215568612

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