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Properties

Label 2-171-171.112-c1-0-12
Degree $2$
Conductor $171$
Sign $0.984 + 0.174i$
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  + (1.58 − 0.575i)2-s + (0.864 + 1.50i)3-s + (0.634 − 0.532i)4-s + (0.605 − 3.43i)5-s + (2.22 + 1.87i)6-s + (0.506 − 0.876i)7-s + (−0.985 + 1.70i)8-s + (−1.50 + 2.59i)9-s + (−1.01 − 5.77i)10-s − 4.20·11-s + (1.34 + 0.492i)12-s + (0.962 + 5.45i)13-s + (0.295 − 1.67i)14-s + (5.67 − 2.05i)15-s + (−0.863 + 4.89i)16-s + (0.762 − 4.32i)17-s + ⋯
L(s)  = 1  + (1.11 − 0.406i)2-s + (0.499 + 0.866i)3-s + (0.317 − 0.266i)4-s + (0.270 − 1.53i)5-s + (0.910 + 0.765i)6-s + (0.191 − 0.331i)7-s + (−0.348 + 0.603i)8-s + (−0.501 + 0.864i)9-s + (−0.322 − 1.82i)10-s − 1.26·11-s + (0.388 + 0.142i)12-s + (0.266 + 1.51i)13-s + (0.0789 − 0.448i)14-s + (1.46 − 0.531i)15-s + (−0.215 + 1.22i)16-s + (0.184 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.984 + 0.174i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07285 - 0.181970i\)
\(L(\frac12)\) \(\approx\) \(2.07285 - 0.181970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.864 - 1.50i)T \)
19 \( 1 + (-1.41 + 4.12i)T \)
good2 \( 1 + (-1.58 + 0.575i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.605 + 3.43i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.506 + 0.876i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.20T + 11T^{2} \)
13 \( 1 + (-0.962 - 5.45i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.762 + 4.32i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (3.97 - 3.33i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.96 + 4.17i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 - 0.544T + 37T^{2} \)
41 \( 1 + (-10.2 + 3.73i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.827 - 0.694i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.66 + 1.39i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.42 + 0.520i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.236 - 0.198i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.165 - 0.938i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.84 + 0.672i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.89 + 2.87i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-3.21 - 2.69i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (0.204 - 1.16i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.37 - 4.11i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.05 - 4.24i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-6.10 + 2.22i)T + (74.3 - 62.3i)T^{2} \)
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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

−12.96017288877480004346480663438, −11.88719134833130433485650847460, −11.01372402714922340501020686085, −9.550383163274792628855299694293, −8.909164046136765516641630928064, −7.79631825606125615842166090124, −5.55680981575590752805275363161, −4.80156875037562347693955292282, −4.08213375639749757962580727693, −2.41430025069113138183212763472, 2.61076203953369416180866434192, 3.51771162627696510881269006721, 5.62160605640971923522145974676, 6.18687269684756583991768064465, 7.40746273388985062648156402989, 8.229632974107244048222300451708, 10.03017978823553589839459756239, 10.81225252019292493843904800358, 12.39068634429839435924132756994, 12.89394026995356710221610927645

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