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Properties

Label 2-171-171.110-c1-0-9
Degree $2$
Conductor $171$
Sign $0.978 + 0.208i$
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  + (−1.50 + 1.26i)2-s + (−1.01 + 1.40i)3-s + (0.325 − 1.84i)4-s + (1.48 − 4.08i)5-s + (−0.249 − 3.40i)6-s + (−0.0709 − 0.122i)7-s + (−0.122 − 0.212i)8-s + (−0.946 − 2.84i)9-s + (2.92 + 8.03i)10-s − 3.16i·11-s + (2.26 + 2.32i)12-s + (−0.419 − 1.15i)13-s + (0.262 + 0.0955i)14-s + (4.22 + 6.22i)15-s + (3.97 + 1.44i)16-s + (−0.619 + 1.70i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.894i)2-s + (−0.585 + 0.811i)3-s + (0.162 − 0.923i)4-s + (0.664 − 1.82i)5-s + (−0.101 − 1.38i)6-s + (−0.0268 − 0.0464i)7-s + (−0.0433 − 0.0751i)8-s + (−0.315 − 0.948i)9-s + (0.925 + 2.54i)10-s − 0.953i·11-s + (0.653 + 0.672i)12-s + (−0.116 − 0.320i)13-s + (0.0701 + 0.0255i)14-s + (1.09 + 1.60i)15-s + (0.994 + 0.362i)16-s + (−0.150 + 0.412i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.978 + 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531814 - 0.0560627i\)
\(L(\frac12)\) \(\approx\) \(0.531814 - 0.0560627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.01 - 1.40i)T \)
19 \( 1 + (-3.16 - 3.00i)T \)
good2 \( 1 + (1.50 - 1.26i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-1.48 + 4.08i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.0709 + 0.122i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.16iT - 11T^{2} \)
13 \( 1 + (0.419 + 1.15i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.619 - 1.70i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.14 + 0.377i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.884 + 5.01i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 8.26iT - 31T^{2} \)
37 \( 1 + 0.413iT - 37T^{2} \)
41 \( 1 + (3.55 - 2.98i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.32 - 7.52i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-12.2 - 2.15i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (0.614 + 0.515i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.650 - 3.68i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-6.52 + 2.37i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.01 - 2.40i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (5.89 - 4.94i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.787 + 4.46i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.163 + 0.450i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.22 - 4.16i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.225 - 1.28i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-6.74 - 8.03i)T + (-16.8 + 95.5i)T^{2} \)
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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

−12.66041537058414211902663227145, −11.68126917286533248771058662889, −10.16275016856737730513401292808, −9.550861532937960401388606520574, −8.713503558422251763682068999911, −7.966616537768778957132828765822, −6.04568878248461474815009197386, −5.59195057549780264433077478614, −4.12559373201604945240603738937, −0.77536884803519970325729822262, 1.89232194182549239594525362153, 2.88260646816468109543264994802, 5.50631155605047012096596790673, 6.89006991864210885560874478329, 7.37466052102299208700855780010, 9.033549113835358202276326587499, 10.16640909535098381703194766946, 10.66512368847289712670533765926, 11.55964487797271432963937574249, 12.32615010054702361372957040197

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