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Properties

Label 2-171-171.14-c1-0-16
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: $\chi_{171} (14, \cdot )$
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

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

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