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Properties

Label 2-171-171.14-c1-0-9
Degree $2$
Conductor $171$
Sign $0.128 + 0.991i$
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  + (−1.28 − 1.07i)2-s + (1.71 − 0.263i)3-s + (0.138 + 0.785i)4-s + (0.287 + 0.788i)5-s + (−2.47 − 1.50i)6-s + (1.60 − 2.77i)7-s + (−1.00 + 1.74i)8-s + (2.86 − 0.902i)9-s + (0.480 − 1.31i)10-s − 0.779i·11-s + (0.443 + 1.30i)12-s + (0.150 − 0.414i)13-s + (−5.02 + 1.83i)14-s + (0.699 + 1.27i)15-s + (4.65 − 1.69i)16-s + (0.0888 + 0.244i)17-s + ⋯
L(s)  = 1  + (−0.905 − 0.760i)2-s + (0.988 − 0.152i)3-s + (0.0692 + 0.392i)4-s + (0.128 + 0.352i)5-s + (−1.01 − 0.613i)6-s + (0.604 − 1.04i)7-s + (−0.355 + 0.615i)8-s + (0.953 − 0.300i)9-s + (0.151 − 0.417i)10-s − 0.235i·11-s + (0.128 + 0.377i)12-s + (0.0418 − 0.114i)13-s + (−1.34 + 0.489i)14-s + (0.180 + 0.329i)15-s + (1.16 − 0.424i)16-s + (0.0215 + 0.0591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.128 + 0.991i$
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: \((2,\ 171,\ (\ :1/2),\ 0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.777122 - 0.683068i\)
\(L(\frac12)\) \(\approx\) \(0.777122 - 0.683068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.263i)T \)
19 \( 1 + (4.35 - 0.225i)T \)
good2 \( 1 + (1.28 + 1.07i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-0.287 - 0.788i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.60 + 2.77i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.779iT - 11T^{2} \)
13 \( 1 + (-0.150 + 0.414i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.0888 - 0.244i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (5.41 - 0.954i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.519 + 2.94i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 - 6.02iT - 31T^{2} \)
37 \( 1 - 11.1iT - 37T^{2} \)
41 \( 1 + (-6.93 - 5.81i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.51 - 8.56i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.169 + 0.0298i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.860 - 0.721i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-0.264 + 1.50i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (3.82 + 1.39i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (4.63 + 5.51i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.53 - 6.32i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.479 + 2.71i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.20 + 3.30i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (10.2 + 5.90i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.74 - 9.86i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (7.42 - 8.84i)T + (-16.8 - 95.5i)T^{2} \)
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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

−12.42423798682596315471573283639, −11.16907924241938808094997484904, −10.38481681500473022730402672126, −9.685775054683811974343738298826, −8.433770125752790618672411810490, −7.87326491239901705777677083586, −6.46572171391188826569986295285, −4.41400723963763703837540727025, −2.89468709184597404934929994242, −1.45351789121764884643333449734, 2.18429720139252723184687022972, 4.05715447861261799141661771841, 5.67494949646567693900864860163, 7.12357684891461436797420829768, 8.108409299451235381497552106565, 8.843595186897472916439531011050, 9.390875488870580615277484700934, 10.63123311323913132532607382716, 12.22439830333384681447446536611, 12.93929026090630094108477509949

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