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Properties

Label 2-171-171.110-c1-0-10
Degree $2$
Conductor $171$
Sign $0.951 + 0.308i$
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  + (−2.02 + 1.69i)2-s + (1.43 − 0.962i)3-s + (0.865 − 4.90i)4-s + (−0.259 + 0.712i)5-s + (−1.28 + 4.39i)6-s + (−1.63 − 2.82i)7-s + (3.94 + 6.83i)8-s + (1.14 − 2.77i)9-s + (−0.685 − 1.88i)10-s − 6.35i·11-s + (−3.47 − 7.90i)12-s + (0.295 + 0.812i)13-s + (8.09 + 2.94i)14-s + (0.312 + 1.27i)15-s + (−10.2 − 3.71i)16-s + (−0.818 + 2.24i)17-s + ⋯
L(s)  = 1  + (−1.43 + 1.20i)2-s + (0.831 − 0.555i)3-s + (0.432 − 2.45i)4-s + (−0.116 + 0.318i)5-s + (−0.522 + 1.79i)6-s + (−0.616 − 1.06i)7-s + (1.39 + 2.41i)8-s + (0.382 − 0.924i)9-s + (−0.216 − 0.595i)10-s − 1.91i·11-s + (−1.00 − 2.28i)12-s + (0.0819 + 0.225i)13-s + (2.16 + 0.787i)14-s + (0.0807 + 0.329i)15-s + (−2.55 − 0.929i)16-s + (−0.198 + 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.951 + 0.308i$
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: \((2,\ 171,\ (\ :1/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672309 - 0.106121i\)
\(L(\frac12)\) \(\approx\) \(0.672309 - 0.106121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.43 + 0.962i)T \)
19 \( 1 + (-4.31 - 0.649i)T \)
good2 \( 1 + (2.02 - 1.69i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.259 - 0.712i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.63 + 2.82i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 6.35iT - 11T^{2} \)
13 \( 1 + (-0.295 - 0.812i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.818 - 2.24i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.245 - 0.0432i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.803 - 4.55i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 1.49iT - 31T^{2} \)
37 \( 1 + 4.34iT - 37T^{2} \)
41 \( 1 + (-1.48 + 1.24i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.639 - 3.62i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (7.25 + 1.27i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-7.32 - 6.14i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.645 + 3.66i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (5.85 - 2.13i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.22 - 1.45i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.638 - 0.536i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.84 - 10.4i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (4.52 - 12.4i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.20 + 3.58i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.93 + 16.6i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-7.16 - 8.54i)T + (-16.8 + 95.5i)T^{2} \)
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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

−13.17540009117842543337854158639, −11.23506981532774528340806568067, −10.36513815290048561583319827309, −9.275136971840450845772634781174, −8.523584756540126612111964193544, −7.56367304875898866757948482950, −6.82928673836230745894635341707, −5.88801781698571107585458883731, −3.40828810729641476398559362052, −0.978352298959568616332738903234, 2.10053304619841469228837333858, 3.10862024862813767704617679355, 4.67491538701041632795290641872, 7.16412005542961406288232511222, 8.164208492368561963448489455192, 9.194213046151587076831759159213, 9.608199331442356168079118185972, 10.40087470491476581794239535989, 11.78969999420474869544315099414, 12.42678682645373707321231573725

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