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Properties

Label 2-171-171.14-c1-0-6
Degree $2$
Conductor $171$
Sign $0.0985 - 0.995i$
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 + (0.658 + 1.60i)3-s + (0.138 + 0.785i)4-s + (−0.199 − 0.548i)5-s + (−0.878 + 2.76i)6-s + (−1.25 + 2.17i)7-s + (1.00 − 1.74i)8-s + (−2.13 + 2.10i)9-s + (0.333 − 0.917i)10-s − 0.404i·11-s + (−1.16 + 0.738i)12-s + (2.33 − 6.40i)13-s + (−3.95 + 1.43i)14-s + (0.747 − 0.681i)15-s + (4.65 − 1.69i)16-s + (1.57 + 4.32i)17-s + ⋯
L(s)  = 1  + (0.905 + 0.760i)2-s + (0.380 + 0.924i)3-s + (0.0692 + 0.392i)4-s + (−0.0892 − 0.245i)5-s + (−0.358 + 1.12i)6-s + (−0.475 + 0.823i)7-s + (0.355 − 0.615i)8-s + (−0.710 + 0.703i)9-s + (0.105 − 0.290i)10-s − 0.121i·11-s + (−0.336 + 0.213i)12-s + (0.646 − 1.77i)13-s + (−1.05 + 0.384i)14-s + (0.192 − 0.175i)15-s + (1.16 − 0.424i)16-s + (0.381 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0985 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.0985 - 0.995i$
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: \((2,\ 171,\ (\ :1/2),\ 0.0985 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37329 + 1.24398i\)
\(L(\frac12)\) \(\approx\) \(1.37329 + 1.24398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.658 - 1.60i)T \)
19 \( 1 + (3.47 + 2.63i)T \)
good2 \( 1 + (-1.28 - 1.07i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (0.199 + 0.548i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.25 - 2.17i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.404iT - 11T^{2} \)
13 \( 1 + (-2.33 + 6.40i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.57 - 4.32i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (6.06 - 1.06i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.264 - 1.49i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 - 2.99iT - 31T^{2} \)
37 \( 1 + 7.01iT - 37T^{2} \)
41 \( 1 + (-3.94 - 3.31i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.105 - 0.598i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (4.36 - 0.770i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.63 - 2.20i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (1.31 - 7.46i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (10.7 + 3.91i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.54 - 1.83i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-5.10 - 4.28i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.34 - 13.3i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.472 + 1.29i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (11.3 + 6.57i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.53 + 8.71i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.35 - 2.80i)T + (-16.8 - 95.5i)T^{2} \)
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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

−13.03297528240045966205415959656, −12.50738208927948665851754845971, −10.79288787514571947484759237667, −10.06297151961346782671872880880, −8.780353969505727907316208909486, −7.924478187614130485152579417817, −6.12018928065128583834817851136, −5.53466843670101034658905479857, −4.26975440490490606662714995763, −3.09868132738771406845118059381, 1.88605080329810382884964646949, 3.34583997408260978210107272394, 4.35878685341890653577164849549, 6.20609984365359357271175089076, 7.17284841274801285593233438715, 8.301370324633776939751821241914, 9.623676650190191612030550509280, 10.98286460907945645674724929740, 11.80422351226932264886624386885, 12.53756032971390189259608201933

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