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Properties

Label 2-171-171.14-c1-0-4
Degree $2$
Conductor $171$
Sign $-0.496 + 0.868i$
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  + (−1.86 − 1.56i)2-s + (−1.40 + 1.01i)3-s + (0.684 + 3.87i)4-s + (0.0603 + 0.165i)5-s + (4.21 + 0.303i)6-s + (0.340 − 0.588i)7-s + (2.36 − 4.09i)8-s + (0.939 − 2.84i)9-s + (0.146 − 0.403i)10-s − 5.07i·11-s + (−4.89 − 4.75i)12-s + (−0.848 + 2.33i)13-s + (−1.55 + 0.566i)14-s + (−0.252 − 0.171i)15-s + (−3.41 + 1.24i)16-s + (−1.07 − 2.95i)17-s + ⋯
L(s)  = 1  + (−1.32 − 1.10i)2-s + (−0.810 + 0.586i)3-s + (0.342 + 1.93i)4-s + (0.0269 + 0.0741i)5-s + (1.71 + 0.123i)6-s + (0.128 − 0.222i)7-s + (0.835 − 1.44i)8-s + (0.313 − 0.949i)9-s + (0.0464 − 0.127i)10-s − 1.52i·11-s + (−1.41 − 1.37i)12-s + (−0.235 + 0.646i)13-s + (−0.416 + 0.151i)14-s + (−0.0652 − 0.0442i)15-s + (−0.854 + 0.311i)16-s + (−0.261 − 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.496 + 0.868i$
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: \((2,\ 171,\ (\ :1/2),\ -0.496 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193923 - 0.334122i\)
\(L(\frac12)\) \(\approx\) \(0.193923 - 0.334122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.40 - 1.01i)T \)
19 \( 1 + (0.122 + 4.35i)T \)
good2 \( 1 + (1.86 + 1.56i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-0.0603 - 0.165i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.340 + 0.588i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.07iT - 11T^{2} \)
13 \( 1 + (0.848 - 2.33i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.07 + 2.95i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (0.413 - 0.0728i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.0433 - 0.245i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + 9.69iT - 31T^{2} \)
37 \( 1 + 5.05iT - 37T^{2} \)
41 \( 1 + (-8.91 - 7.48i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.329 + 1.86i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.352 + 0.0621i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (4.97 - 4.17i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.574 - 3.25i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (7.26 + 2.64i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (4.89 + 5.83i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (10.2 + 8.56i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.55 - 14.4i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.04 - 5.61i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.844 + 0.487i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.48 + 8.40i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-8.73 + 10.4i)T + (-16.8 - 95.5i)T^{2} \)
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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

−11.78144839240413789584676646943, −11.19997961892020366922926702875, −10.64635962347737243580488985825, −9.453260563393060053447497589117, −8.899395922339049933060582647526, −7.51264850693446009845468795418, −6.09475660967961523334554216699, −4.37235876670099440361746848143, −2.85959993701283800515640726211, −0.63801981691009384345949693835, 1.61202404851202775768854012557, 4.95085069437632087397836249225, 6.01709945459651121279465996065, 7.04639421125818055735995652675, 7.74159969131299684694385246220, 8.811734052246018755783881052994, 10.08518936812536544329927555064, 10.63978454178989736008884830713, 12.15110566172979926152007742560, 12.85504131036455083538718259039

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