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Properties

Label 2-171-57.8-c1-0-6
Degree $2$
Conductor $171$
Sign $-0.672 + 0.740i$
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  + (1.13 − 1.97i)2-s + (−1.59 − 2.76i)4-s + (−2.90 − 1.67i)5-s + 3.54·7-s − 2.71·8-s + (−6.61 + 3.81i)10-s + 0.251i·11-s + (−4.29 + 2.47i)13-s + (4.04 − 6.99i)14-s + (0.0992 − 0.171i)16-s + (3.11 + 1.80i)17-s + (4.15 − 1.30i)19-s + 10.6i·20-s + (0.496 + 0.286i)22-s + (3.89 − 2.24i)23-s + ⋯
L(s)  = 1  + (0.805 − 1.39i)2-s + (−0.797 − 1.38i)4-s + (−1.29 − 0.749i)5-s + 1.34·7-s − 0.959·8-s + (−2.09 + 1.20i)10-s + 0.0758i·11-s + (−1.18 + 0.687i)13-s + (1.07 − 1.87i)14-s + (0.0248 − 0.0429i)16-s + (0.756 + 0.436i)17-s + (0.954 − 0.298i)19-s + 2.39i·20-s + (0.105 + 0.0610i)22-s + (0.811 − 0.468i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.672 + 0.740i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.672 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613386 - 1.38621i\)
\(L(\frac12)\) \(\approx\) \(0.613386 - 1.38621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.15 + 1.30i)T \)
good2 \( 1 + (-1.13 + 1.97i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.90 + 1.67i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 - 0.251iT - 11T^{2} \)
13 \( 1 + (4.29 - 2.47i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.11 - 1.80i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.89 + 2.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.27 - 3.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.28iT - 31T^{2} \)
37 \( 1 + 6.47iT - 37T^{2} \)
41 \( 1 + (4.96 - 8.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.38 - 5.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.96 - 5.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.217 + 0.377i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.06 + 3.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.46 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.273 + 0.157i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.10 - 7.11i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.356 + 0.616i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.57 + 4.37i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.31iT - 83T^{2} \)
89 \( 1 + (1.20 + 2.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.36 - 5.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.18032290203935155778558536919, −11.63334069844174737593427247682, −10.93676620042951365900003336177, −9.643271665500291275969222692455, −8.347899484387306616524268240924, −7.38898953137667737372131481762, −4.85963636875282064681027908067, −4.75781177852137440189233310067, −3.25386922790423552262060965944, −1.41966994440504079027652531164, 3.32871028381778559666733159906, 4.65298143960941503120942955689, 5.47790576849287125286758892012, 7.16019719556810628386134554877, 7.59896239307764555696743419206, 8.330873281329097599861394838966, 10.21509128366126033483695516967, 11.54773949405570145009276578409, 12.07176009373414844355936943646, 13.53697734871859099902799117964

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