Properties

Label 2-171-171.110-c1-0-3
Degree $2$
Conductor $171$
Sign $-0.260 - 0.965i$
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  + (−0.907 + 0.761i)2-s + (0.509 − 1.65i)3-s + (−0.103 + 0.588i)4-s + (−1.20 + 3.32i)5-s + (0.797 + 1.88i)6-s + (0.600 + 1.04i)7-s + (−1.53 − 2.66i)8-s + (−2.48 − 1.68i)9-s + (−1.43 − 3.93i)10-s + 3.37i·11-s + (0.921 + 0.471i)12-s + (1.06 + 2.92i)13-s + (−1.33 − 0.486i)14-s + (4.88 + 3.69i)15-s + (2.30 + 0.837i)16-s + (−1.91 + 5.24i)17-s + ⋯
L(s)  = 1  + (−0.641 + 0.538i)2-s + (0.294 − 0.955i)3-s + (−0.0518 + 0.294i)4-s + (−0.540 + 1.48i)5-s + (0.325 + 0.771i)6-s + (0.227 + 0.393i)7-s + (−0.543 − 0.941i)8-s + (−0.826 − 0.562i)9-s + (−0.453 − 1.24i)10-s + 1.01i·11-s + (0.265 + 0.136i)12-s + (0.295 + 0.810i)13-s + (−0.357 − 0.130i)14-s + (1.26 + 0.954i)15-s + (0.575 + 0.209i)16-s + (−0.463 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.260 - 0.965i$
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: \((2,\ 171,\ (\ :1/2),\ -0.260 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446371 + 0.583021i\)
\(L(\frac12)\) \(\approx\) \(0.446371 + 0.583021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.509 + 1.65i)T \)
19 \( 1 + (-3.51 + 2.57i)T \)
good2 \( 1 + (0.907 - 0.761i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.20 - 3.32i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.600 - 1.04i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.37iT - 11T^{2} \)
13 \( 1 + (-1.06 - 2.92i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.91 - 5.24i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.71 - 0.479i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.75 + 9.93i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 0.427iT - 31T^{2} \)
37 \( 1 + 0.0841iT - 37T^{2} \)
41 \( 1 + (1.55 - 1.30i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.0600 + 0.340i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-7.15 - 1.26i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (4.91 + 4.12i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.479 - 2.71i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-4.68 + 1.70i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (4.40 - 5.25i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-10.6 + 8.94i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-2.30 - 13.0i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (5.23 - 14.3i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-9.41 + 5.43i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.172 - 0.976i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.27 + 2.71i)T + (-16.8 + 95.5i)T^{2} \)
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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

−13.03319090747087097381278669515, −11.97541570094268394037994782808, −11.26910730721708126867770562357, −9.813970511008913837184098609407, −8.666512081195970839426785436283, −7.72234610289217912244281683929, −6.99563481853313714030130355287, −6.31117554980177475478495396935, −3.84744174477944293881614676412, −2.43807473500303177502030308082, 0.854407424032179166530072191532, 3.25009320401758096458039573191, 4.81863748914292214632504121745, 5.50182540291232594483824618590, 7.88423215658436029650622072864, 8.798704652896941384625621432548, 9.246881787889943961793972993962, 10.49341944881524843787152308001, 11.19233292896659173920612936224, 12.16535799090382865708410121603

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