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Properties

Label 2-171-19.7-c1-0-1
Degree $2$
Conductor $171$
Sign $-0.567 - 0.823i$
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  + (1.16 + 2.02i)2-s + (−1.72 + 2.98i)4-s + (0.524 + 0.908i)5-s − 3.44·7-s − 3.38·8-s + (−1.22 + 2.12i)10-s + 5.71·11-s + (0.5 − 0.866i)13-s + (−4.02 − 6.97i)14-s + (−0.500 − 0.866i)16-s + (−1.04 − 1.81i)17-s + (1 − 4.24i)19-s − 3.61·20-s + (6.67 + 11.5i)22-s + (−1.80 + 3.13i)23-s + ⋯
L(s)  = 1  + (0.825 + 1.42i)2-s + (−0.862 + 1.49i)4-s + (0.234 + 0.406i)5-s − 1.30·7-s − 1.19·8-s + (−0.387 + 0.670i)10-s + 1.72·11-s + (0.138 − 0.240i)13-s + (−1.07 − 1.86i)14-s + (−0.125 − 0.216i)16-s + (−0.254 − 0.440i)17-s + (0.229 − 0.973i)19-s − 0.809·20-s + (1.42 + 2.46i)22-s + (−0.377 + 0.653i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753362 + 1.43317i\)
\(L(\frac12)\) \(\approx\) \(0.753362 + 1.43317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-1 + 4.24i)T \)
good2 \( 1 + (-1.16 - 2.02i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.524 - 0.908i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 - 5.71T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.04 + 1.81i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.80 - 3.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.61 + 6.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.44T + 31T^{2} \)
37 \( 1 - 3.89T + 37T^{2} \)
41 \( 1 + (-4.66 - 8.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.17 + 5.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.66 - 8.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.524 - 0.908i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.90 + 6.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.174 - 0.301i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.61 + 6.26i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.174 + 0.301i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (-2.62 + 4.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.55 + 2.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.40140601187099439317413135446, −12.54798826804145465692591500639, −11.36140530109858236375060029902, −9.763344036572334146425231920320, −8.944538258651007672583144140959, −7.44647373838757797092132209397, −6.51947197923179258827423026222, −6.05774345936676534629685546238, −4.45028810441098974392498080797, −3.27209989116301653810870929278, 1.55527538572307297230851856321, 3.33908773496856145759819457692, 4.15500534839769717008787518503, 5.69932283889306553982380002765, 6.79775061315552148733203431657, 8.929230373849495247194570113950, 9.610582587923904372606763061310, 10.58775125566395322707918686422, 11.66587319075976444966662476591, 12.53330189331524522852728494892

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