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Properties

Label 2-171-171.110-c1-0-5
Degree $2$
Conductor $171$
Sign $-0.263 - 0.964i$
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  + (−0.247 + 0.207i)2-s + (0.668 + 1.59i)3-s + (−0.329 + 1.86i)4-s + (−0.0294 + 0.0810i)5-s + (−0.496 − 0.256i)6-s + (−0.0754 − 0.130i)7-s + (−0.628 − 1.08i)8-s + (−2.10 + 2.13i)9-s + (−0.00951 − 0.0261i)10-s − 1.72i·11-s + (−3.20 + 0.721i)12-s + (0.805 + 2.21i)13-s + (0.0457 + 0.0166i)14-s + (−0.149 + 0.00700i)15-s + (−3.18 − 1.15i)16-s + (0.788 − 2.16i)17-s + ⋯
L(s)  = 1  + (−0.174 + 0.146i)2-s + (0.385 + 0.922i)3-s + (−0.164 + 0.933i)4-s + (−0.0131 + 0.0362i)5-s + (−0.202 − 0.104i)6-s + (−0.0285 − 0.0494i)7-s + (−0.222 − 0.384i)8-s + (−0.702 + 0.711i)9-s + (−0.00300 − 0.00826i)10-s − 0.520i·11-s + (−0.924 + 0.208i)12-s + (0.223 + 0.614i)13-s + (0.0122 + 0.00445i)14-s + (−0.0385 + 0.00180i)15-s + (−0.795 − 0.289i)16-s + (0.191 − 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.263 - 0.964i$
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: \((2,\ 171,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666356 + 0.873028i\)
\(L(\frac12)\) \(\approx\) \(0.666356 + 0.873028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.668 - 1.59i)T \)
19 \( 1 + (-3.17 - 2.98i)T \)
good2 \( 1 + (0.247 - 0.207i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.0294 - 0.0810i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.0754 + 0.130i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.72iT - 11T^{2} \)
13 \( 1 + (-0.805 - 2.21i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.788 + 2.16i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.13 - 0.905i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.930 + 5.27i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 0.157iT - 31T^{2} \)
37 \( 1 - 1.03iT - 37T^{2} \)
41 \( 1 + (-3.23 + 2.71i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.868 + 4.92i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (11.1 + 1.95i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.60 - 3.02i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.485 + 2.75i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.10 + 1.85i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.30 + 1.55i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (10.6 - 8.95i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.405 + 2.30i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.22 - 3.36i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.67 - 2.12i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.82 - 16.0i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.33 - 1.59i)T + (-16.8 + 95.5i)T^{2} \)
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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

−13.23654326934959752512448291896, −11.89002842447567224087786631423, −11.11568159897347726512456860389, −9.805195476199172357366524945289, −8.982907816581127638761675129622, −8.137242725749817936981079609075, −6.99288804700337184430461078164, −5.30847819511317405826584591488, −3.97592262116071246236790573839, −2.97519450572684823750549477273, 1.20253783448205833670285456604, 2.86796939479369088918710575666, 4.94859418575365115147645776123, 6.17738466545435547444984256666, 7.20819382752546552331792465694, 8.481283040310184466433388996538, 9.337806069461281236220201684929, 10.47719224275059171907936406384, 11.46255629181612395761151017968, 12.65229992838771359727358366980

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