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Properties

Label 2-171-171.106-c1-0-9
Degree $2$
Conductor $171$
Sign $0.985 + 0.169i$
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  + (−0.395 − 0.685i)2-s + (0.659 + 1.60i)3-s + (0.686 − 1.18i)4-s + 2.59·5-s + (0.837 − 1.08i)6-s + (−0.373 + 0.646i)7-s − 2.67·8-s + (−2.13 + 2.11i)9-s + (−1.02 − 1.77i)10-s + (−1.28 + 2.23i)11-s + (2.35 + 0.315i)12-s + (3.09 − 5.35i)13-s + 0.590·14-s + (1.70 + 4.14i)15-s + (−0.315 − 0.546i)16-s + (0.119 − 0.207i)17-s + ⋯
L(s)  = 1  + (−0.279 − 0.484i)2-s + (0.380 + 0.924i)3-s + (0.343 − 0.594i)4-s + 1.15·5-s + (0.341 − 0.443i)6-s + (−0.141 + 0.244i)7-s − 0.944·8-s + (−0.710 + 0.703i)9-s + (−0.324 − 0.561i)10-s + (−0.388 + 0.672i)11-s + (0.680 + 0.0911i)12-s + (0.858 − 1.48i)13-s + 0.157·14-s + (0.440 + 1.07i)15-s + (−0.0788 − 0.136i)16-s + (0.0291 − 0.0504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32389 - 0.113169i\)
\(L(\frac12)\) \(\approx\) \(1.32389 - 0.113169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.659 - 1.60i)T \)
19 \( 1 + (-3.89 - 1.95i)T \)
good2 \( 1 + (0.395 + 0.685i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 2.59T + 5T^{2} \)
7 \( 1 + (0.373 - 0.646i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.28 - 2.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.09 + 5.35i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.119 + 0.207i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.93 - 3.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.79T + 29T^{2} \)
31 \( 1 + (3.77 + 6.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 - 8.15T + 41T^{2} \)
43 \( 1 + (-1.44 - 2.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.52T + 47T^{2} \)
53 \( 1 + (5.57 + 9.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.344T + 59T^{2} \)
61 \( 1 - 0.0790T + 61T^{2} \)
67 \( 1 + (4.61 - 7.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.15 + 3.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.63 + 2.82i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.57 - 6.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.78 + 3.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.21 - 9.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.20 - 2.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.79687999453057032943760338971, −11.34766133503159902440279325427, −10.49055411864856685485065336410, −9.779030717064741830912777663629, −9.244044646972639526153012512524, −7.78713167872614761558762152567, −5.84331057471553941280471185729, −5.43352652436990662067922865038, −3.34098345014464191789167403189, −2.02111102374708899310869295638, 1.95682294504981935436964407910, 3.41079853015847432658914766786, 5.78630243014799546697249862588, 6.59720801733243397902463702115, 7.47233316951441998837005575514, 8.732089901467421455345160288052, 9.272190597476128013432112450572, 10.92148081377030961061609655233, 11.93471836719889990382803022884, 12.97772842025074563663931467926

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