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Properties

Label 2-171-171.110-c1-0-1
Degree $2$
Conductor $171$
Sign $-0.667 + 0.744i$
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  + (−1.80 + 1.51i)2-s + (−0.419 + 1.68i)3-s + (0.621 − 3.52i)4-s + (−1.22 + 3.36i)5-s + (−1.79 − 3.67i)6-s + (0.850 + 1.47i)7-s + (1.86 + 3.22i)8-s + (−2.64 − 1.40i)9-s + (−2.89 − 7.94i)10-s − 0.0685i·11-s + (5.65 + 2.52i)12-s + (0.947 + 2.60i)13-s + (−3.77 − 1.37i)14-s + (−5.14 − 3.46i)15-s + (−1.54 − 0.560i)16-s + (2.40 − 6.59i)17-s + ⋯
L(s)  = 1  + (−1.27 + 1.07i)2-s + (−0.241 + 0.970i)3-s + (0.310 − 1.76i)4-s + (−0.547 + 1.50i)5-s + (−0.731 − 1.50i)6-s + (0.321 + 0.557i)7-s + (0.658 + 1.14i)8-s + (−0.882 − 0.469i)9-s + (−0.914 − 2.51i)10-s − 0.0206i·11-s + (1.63 + 0.727i)12-s + (0.262 + 0.722i)13-s + (−1.00 − 0.367i)14-s + (−1.32 − 0.895i)15-s + (−0.385 − 0.140i)16-s + (0.582 − 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.667 + 0.744i$
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: \((2,\ 171,\ (\ :1/2),\ -0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169068 - 0.378535i\)
\(L(\frac12)\) \(\approx\) \(0.169068 - 0.378535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.419 - 1.68i)T \)
19 \( 1 + (3.88 + 1.97i)T \)
good2 \( 1 + (1.80 - 1.51i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.22 - 3.36i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.850 - 1.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.0685iT - 11T^{2} \)
13 \( 1 + (-0.947 - 2.60i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.40 + 6.59i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.95 - 0.521i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (1.00 - 5.69i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 - 2.56iT - 31T^{2} \)
37 \( 1 - 7.77iT - 37T^{2} \)
41 \( 1 + (8.32 - 6.98i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.683 - 3.87i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.713 + 0.125i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-6.47 - 5.43i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (1.93 + 10.9i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (1.83 - 0.666i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (5.17 - 6.16i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.92 + 4.13i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.28 - 7.28i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-3.66 + 10.0i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.302 - 0.174i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.881 - 5.00i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-2.44 - 2.91i)T + (-16.8 + 95.5i)T^{2} \)
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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

−14.05521878667498806678738903909, −11.76265142085014642950713873126, −11.07900213268965368492405618060, −10.20409980232015100748881563335, −9.273632403120324975769974204530, −8.377500829339848531194175325450, −7.11808011354719171620269443356, −6.42428355577325082031708981999, −5.01839647959743136966768725022, −3.12404876114490107724517224723, 0.61532217046761113072306906505, 1.79787074278136269155466358964, 3.90031484082019796371676300379, 5.67271866179186160109526756563, 7.56337635050478167702189049038, 8.262056083322919421244794380278, 8.812952979931391422452329443163, 10.31255084467421385129037277557, 11.08869210257638181451987220271, 12.22358786250760377950140527916

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