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Properties

Label 2-171-19.7-c1-0-4
Degree $2$
Conductor $171$
Sign $0.910 - 0.412i$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + 7-s + 3·8-s + 2·11-s + (−2.5 + 4.33i)13-s + (0.5 + 0.866i)14-s + (0.500 + 0.866i)16-s + (−2 − 3.46i)17-s + (−4 − 1.73i)19-s + (1 + 1.73i)22-s + (−2 + 3.46i)23-s + (2.5 − 4.33i)25-s − 5·26-s + (0.500 − 0.866i)28-s + (−4 + 6.92i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + 0.377·7-s + 1.06·8-s + 0.603·11-s + (−0.693 + 1.20i)13-s + (0.133 + 0.231i)14-s + (0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (−0.917 − 0.397i)19-s + (0.213 + 0.369i)22-s + (−0.417 + 0.722i)23-s + (0.5 − 0.866i)25-s − 0.980·26-s + (0.0944 − 0.163i)28-s + (−0.742 + 1.28i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48087 + 0.320057i\)
\(L(\frac12)\) \(\approx\) \(1.48087 + 0.320057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (4 + 1.73i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.00358723600296357930201328921, −11.72597041430198767534386264957, −10.99314825236889749223271038992, −9.770082112684664126323073126712, −8.739122200387657435331285421480, −7.22863667257789865628661188720, −6.62899897096283456553033020916, −5.25237231027046152650155683618, −4.26218036196311968434327440702, −1.98588800280289030643560997342, 2.05930157279150413425232928117, 3.55767824986701375675868417535, 4.73389840297896153033691600548, 6.30003152929570013923168302982, 7.63756970877524069298361787844, 8.465345632336524695421519797279, 9.991703005421745185780464941918, 10.89319498300486686465672884818, 11.73795319515122549705207836049, 12.72990820154975166673138353569

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