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Properties

Label 2-171-171.106-c1-0-12
Degree $2$
Conductor $171$
Sign $0.995 + 0.0983i$
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  + (0.0973 + 0.168i)2-s + (1.55 + 0.767i)3-s + (0.981 − 1.69i)4-s − 1.90·5-s + (0.0217 + 0.336i)6-s + (1.69 − 2.93i)7-s + 0.771·8-s + (1.82 + 2.38i)9-s + (−0.185 − 0.321i)10-s + (−0.311 + 0.539i)11-s + (2.82 − 1.88i)12-s + (−1.84 + 3.19i)13-s + 0.659·14-s + (−2.95 − 1.46i)15-s + (−1.88 − 3.26i)16-s + (−3.04 + 5.27i)17-s + ⋯
L(s)  = 1  + (0.0688 + 0.119i)2-s + (0.896 + 0.443i)3-s + (0.490 − 0.849i)4-s − 0.852·5-s + (0.00886 + 0.137i)6-s + (0.640 − 1.10i)7-s + 0.272·8-s + (0.607 + 0.794i)9-s + (−0.0586 − 0.101i)10-s + (−0.0939 + 0.162i)11-s + (0.816 − 0.544i)12-s + (−0.511 + 0.886i)13-s + 0.176·14-s + (−0.763 − 0.377i)15-s + (−0.471 − 0.817i)16-s + (−0.739 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.995 + 0.0983i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.995 + 0.0983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54249 - 0.0760145i\)
\(L(\frac12)\) \(\approx\) \(1.54249 - 0.0760145i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.55 - 0.767i)T \)
19 \( 1 + (1.14 - 4.20i)T \)
good2 \( 1 + (-0.0973 - 0.168i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.90T + 5T^{2} \)
7 \( 1 + (-1.69 + 2.93i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.311 - 0.539i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.84 - 3.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.04 - 5.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.92 + 6.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.18T + 29T^{2} \)
31 \( 1 + (0.910 + 1.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.63T + 37T^{2} \)
41 \( 1 + 4.03T + 41T^{2} \)
43 \( 1 + (2.54 + 4.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + (-1.93 - 3.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.50T + 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 + (0.523 - 0.905i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.56 + 2.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.06 + 3.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.16 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.35 + 9.27i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.25 + 9.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.34 + 12.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.92187125135635673090444783581, −11.46342756842210731406295119047, −10.66314899808795755620839610989, −9.932106718302559886444197897500, −8.529787757656278665377268428797, −7.61434239699065937296963213000, −6.64359672580377016738176872996, −4.72245160522546396476182276409, −3.96298076877935474919810884192, −1.93111397602633566881154949789, 2.38755040236041694754207931308, 3.34859675423606723252733043629, 4.95352981339171122747367045196, 6.89024995917755070337244159088, 7.75229203216001470271079014632, 8.442252834254829188668519418033, 9.426969491971795461826356873279, 11.37400414603056887289450513294, 11.68121303362098467914217899267, 12.80433009405665120657182920649

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