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Properties

Label 2-171-19.12-c2-0-6
Degree $2$
Conductor $171$
Sign $0.983 - 0.178i$
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  + (0.583 − 0.336i)2-s + (−1.77 + 3.07i)4-s + (−2.27 − 3.93i)5-s + 9.87·7-s + 5.08i·8-s + (−2.65 − 1.53i)10-s + 15.6·11-s + (13.3 + 7.68i)13-s + (5.76 − 3.32i)14-s + (−5.37 − 9.31i)16-s + (12.4 + 21.5i)17-s + (−18.4 − 4.63i)19-s + 16.1·20-s + (9.12 − 5.26i)22-s + (−4.15 + 7.19i)23-s + ⋯
L(s)  = 1  + (0.291 − 0.168i)2-s + (−0.443 + 0.767i)4-s + (−0.454 − 0.787i)5-s + 1.41·7-s + 0.635i·8-s + (−0.265 − 0.153i)10-s + 1.42·11-s + (1.02 + 0.590i)13-s + (0.411 − 0.237i)14-s + (−0.336 − 0.582i)16-s + (0.730 + 1.26i)17-s + (−0.969 − 0.243i)19-s + 0.806·20-s + (0.414 − 0.239i)22-s + (−0.180 + 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.983 - 0.178i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.983 - 0.178i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.74684 + 0.157007i\)
\(L(\frac12)\) \(\approx\) \(1.74684 + 0.157007i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (18.4 + 4.63i)T \)
good2 \( 1 + (-0.583 + 0.336i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (2.27 + 3.93i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 9.87T + 49T^{2} \)
11 \( 1 - 15.6T + 121T^{2} \)
13 \( 1 + (-13.3 - 7.68i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-12.4 - 21.5i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (4.15 - 7.19i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (27.3 + 15.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 30.9iT - 961T^{2} \)
37 \( 1 - 17.3iT - 1.36e3T^{2} \)
41 \( 1 + (-44.2 + 25.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.773 + 1.33i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (5.09 - 8.81i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (4.54 + 2.62i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (68.4 - 39.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (53.9 - 93.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.9 + 9.80i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-45.3 + 26.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (36.6 + 63.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (27.1 - 15.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 42.6T + 6.88e3T^{2} \)
89 \( 1 + (114. + 66.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (70.2 - 40.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.41348140605298236936369565036, −11.72636815232303596507437832560, −10.98086748977877295860602396903, −9.116685212894932215487994443241, −8.483044762553762325971569005543, −7.71155833066377134056235916287, −5.97112151800625340654194573376, −4.37825163417346417460519279224, −3.99083230508453840355131077720, −1.57157107967013314014852328612, 1.33810331491247913736206653133, 3.64079735509608469227193632118, 4.80904916845132666853380906035, 6.02921277364822162061382557859, 7.15967867952649282589927613363, 8.397288083990154984317569720220, 9.445107718834016117420445779315, 10.84168167610591961841552373104, 11.21926542345912141250593644024, 12.47525206836194982495967469868

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