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Properties

Label 2-171-171.14-c1-0-5
Degree $2$
Conductor $171$
Sign $0.721 - 0.692i$
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  + (0.353 + 0.296i)2-s + (0.726 + 1.57i)3-s + (−0.310 − 1.76i)4-s + (0.919 + 2.52i)5-s + (−0.209 + 0.770i)6-s + (2.00 − 3.47i)7-s + (0.873 − 1.51i)8-s + (−1.94 + 2.28i)9-s + (−0.423 + 1.16i)10-s + 3.58i·11-s + (2.54 − 1.76i)12-s + (−1.68 + 4.62i)13-s + (1.73 − 0.633i)14-s + (−3.30 + 3.28i)15-s + (−2.60 + 0.947i)16-s + (−2.40 − 6.59i)17-s + ⋯
L(s)  = 1  + (0.249 + 0.209i)2-s + (0.419 + 0.907i)3-s + (−0.155 − 0.880i)4-s + (0.411 + 1.12i)5-s + (−0.0854 + 0.314i)6-s + (0.758 − 1.31i)7-s + (0.308 − 0.534i)8-s + (−0.648 + 0.761i)9-s + (−0.134 + 0.368i)10-s + 1.08i·11-s + (0.733 − 0.510i)12-s + (−0.466 + 1.28i)13-s + (0.464 − 0.169i)14-s + (−0.853 + 0.847i)15-s + (−0.650 + 0.236i)16-s + (−0.582 − 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.721 - 0.692i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40799 + 0.566747i\)
\(L(\frac12)\) \(\approx\) \(1.40799 + 0.566747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.726 - 1.57i)T \)
19 \( 1 + (0.557 + 4.32i)T \)
good2 \( 1 + (-0.353 - 0.296i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-0.919 - 2.52i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-2.00 + 3.47i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.58iT - 11T^{2} \)
13 \( 1 + (1.68 - 4.62i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.40 + 6.59i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.11 + 0.372i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.213 + 1.21i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + 4.02iT - 31T^{2} \)
37 \( 1 - 0.477iT - 37T^{2} \)
41 \( 1 + (-1.67 - 1.40i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.397 + 2.25i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.61 - 0.989i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.30 - 1.93i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (1.51 - 8.59i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (7.98 + 2.90i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-6.88 - 8.21i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.839 - 0.704i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.773 + 4.38i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-3.11 - 8.55i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-13.1 - 7.60i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.329 + 1.86i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (6.65 - 7.92i)T + (-16.8 - 95.5i)T^{2} \)
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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

−13.61548192789388251364369193279, −11.35661691867139929677021859598, −10.82508608263729507746476582204, −9.854579386328654951668261350578, −9.304003066660635703059527081540, −7.33776350670811612945004131160, −6.74190184174890705291490582360, −4.87551257174389426006864090165, −4.34343105052810336874253988176, −2.34148513782896097680561718056, 1.84986797407846334211864261520, 3.27529353631986578992390218731, 5.10904659379613554671416891903, 6.03960299997308637366386886598, 8.018452384047185341702029547478, 8.363657678044015554411833627514, 9.071287149176142342026759744693, 10.99807067673000720762029483442, 12.17003049599699152073475186147, 12.64063872673087767102411777812

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