Properties

Label 2-171-171.103-c2-0-10
Degree $2$
Conductor $171$
Sign $-0.887 + 0.461i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.89i·2-s + (−2.37 + 1.83i)3-s − 4.39·4-s + (4.55 + 7.88i)5-s + (−5.31 − 6.87i)6-s + (3.39 + 5.88i)7-s − 1.13i·8-s + (2.25 − 8.71i)9-s + (−22.8 + 13.1i)10-s + (4.36 + 7.55i)11-s + (10.4 − 8.06i)12-s − 13.8i·13-s + (−17.0 + 9.84i)14-s + (−25.2 − 10.3i)15-s − 14.2·16-s + (13.7 − 23.8i)17-s + ⋯
L(s)  = 1  + 1.44i·2-s + (−0.790 + 0.612i)3-s − 1.09·4-s + (0.910 + 1.57i)5-s + (−0.886 − 1.14i)6-s + (0.485 + 0.840i)7-s − 0.142i·8-s + (0.250 − 0.968i)9-s + (−2.28 + 1.31i)10-s + (0.396 + 0.687i)11-s + (0.868 − 0.672i)12-s − 1.06i·13-s + (−1.21 + 0.702i)14-s + (−1.68 − 0.689i)15-s − 0.892·16-s + (0.810 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.887 + 0.461i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.331568 - 1.35685i\)
\(L(\frac12)\) \(\approx\) \(0.331568 - 1.35685i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.37 - 1.83i)T \)
19 \( 1 + (-10.5 + 15.8i)T \)
good2 \( 1 - 2.89iT - 4T^{2} \)
5 \( 1 + (-4.55 - 7.88i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.39 - 5.88i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-4.36 - 7.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 13.8iT - 169T^{2} \)
17 \( 1 + (-13.7 + 23.8i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 21.8T + 529T^{2} \)
29 \( 1 + (17.8 + 10.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-20.2 - 11.6i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 22.7iT - 1.36e3T^{2} \)
41 \( 1 + (28.9 - 16.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 3.79T + 1.84e3T^{2} \)
47 \( 1 + (-23.1 + 40.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (34.8 - 20.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-34.4 + 19.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (60.3 - 104. i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 31.1iT - 4.48e3T^{2} \)
71 \( 1 + (-46.4 - 26.8i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (1.89 - 3.27i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 61.4iT - 6.24e3T^{2} \)
83 \( 1 + (-14.5 - 25.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-10.0 + 5.78i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 126. iT - 9.40e3T^{2} \)
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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

−13.52872508943409815728309233891, −11.89278119425585839417936538135, −11.04526932886349590515605530210, −9.945524812415322908252136747067, −9.128131184797572314082685781772, −7.42354511401440575410596759845, −6.74222173904898468079719577284, −5.65806389369321415906921027323, −5.10047425758511273667138766388, −2.85299691717803693911047784644, 1.11487629342762487981184335633, 1.61966090666001447686527293691, 4.03405403736808938132669712402, 5.14957381454950167437065129932, 6.37421024430675089928077668857, 8.068337611396661738375119496482, 9.231757087839204562898532634488, 10.20387803554127747247392342841, 11.11314933223431193246593462828, 12.03809736815124117867066999668

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