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Properties

Label 2-171-171.110-c1-0-17
Degree $2$
Conductor $171$
Sign $-0.424 + 0.905i$
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  + (1.98 − 1.66i)2-s + (−0.713 − 1.57i)3-s + (0.819 − 4.64i)4-s + (−0.441 + 1.21i)5-s + (−4.04 − 1.94i)6-s + (2.16 + 3.74i)7-s + (−3.52 − 6.10i)8-s + (−1.98 + 2.25i)9-s + (1.14 + 3.14i)10-s + 2.72i·11-s + (−7.92 + 2.02i)12-s + (−1.29 − 3.55i)13-s + (10.5 + 3.83i)14-s + (2.23 − 0.169i)15-s + (−8.30 − 3.02i)16-s + (0.00982 − 0.0269i)17-s + ⋯
L(s)  = 1  + (1.40 − 1.17i)2-s + (−0.412 − 0.911i)3-s + (0.409 − 2.32i)4-s + (−0.197 + 0.542i)5-s + (−1.65 − 0.793i)6-s + (0.817 + 1.41i)7-s + (−1.24 − 2.15i)8-s + (−0.660 + 0.751i)9-s + (0.362 + 0.995i)10-s + 0.822i·11-s + (−2.28 + 0.584i)12-s + (−0.359 − 0.987i)13-s + (2.81 + 1.02i)14-s + (0.576 − 0.0437i)15-s + (−2.07 − 0.756i)16-s + (0.00238 − 0.00654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.424 + 0.905i$
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: \((2,\ 171,\ (\ :1/2),\ -0.424 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07508 - 1.69099i\)
\(L(\frac12)\) \(\approx\) \(1.07508 - 1.69099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.713 + 1.57i)T \)
19 \( 1 + (0.236 + 4.35i)T \)
good2 \( 1 + (-1.98 + 1.66i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.441 - 1.21i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-2.16 - 3.74i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.72iT - 11T^{2} \)
13 \( 1 + (1.29 + 3.55i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.00982 + 0.0269i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (5.97 + 1.05i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.592 - 3.35i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 - 7.25iT - 37T^{2} \)
41 \( 1 + (2.90 - 2.43i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.543 + 3.08i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (7.07 + 1.24i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-4.24 - 3.56i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (1.88 + 10.6i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-9.95 + 3.62i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-6.64 + 7.91i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (4.93 - 4.13i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.86 - 10.5i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.76 - 4.86i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.70 - 4.44i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.81 + 10.2i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (0.845 + 1.00i)T + (-16.8 + 95.5i)T^{2} \)
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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

−12.44115334775737824001275167297, −11.65499890247807057306865878406, −11.10936798843308199860817145375, −9.938712594682753627633043183589, −8.238584297944481069611378899553, −6.77174823297798247028146659804, −5.56978337036689272962954494026, −4.85832335020322644656949022876, −2.88065359806123175283508203152, −1.95529027207016422326907919172, 3.80626365783300294641625455862, 4.33423180142113164692504341339, 5.34270592599999670991816940507, 6.46161988222635116305808817171, 7.68084850263016354970411211488, 8.614520039443079548155997227981, 10.25942231786318472738369990331, 11.45450845729989728163747941467, 12.15709455445909574100916041652, 13.46514837534700346438075072783

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