Properties

Label 2-171-171.103-c2-0-19
Degree $2$
Conductor $171$
Sign $0.255 - 0.966i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.07i·2-s + (2.55 − 1.57i)3-s − 0.325·4-s + (3.09 + 5.35i)5-s + (3.26 + 5.31i)6-s + (−2.07 − 3.59i)7-s + 7.64i·8-s + (4.06 − 8.03i)9-s + (−11.1 + 6.42i)10-s + (0.895 + 1.55i)11-s + (−0.831 + 0.511i)12-s − 6.42i·13-s + (7.48 − 4.32i)14-s + (16.3 + 8.82i)15-s − 17.1·16-s + (1.28 − 2.22i)17-s + ⋯
L(s)  = 1  + 1.03i·2-s + (0.851 − 0.523i)3-s − 0.0813·4-s + (0.618 + 1.07i)5-s + (0.544 + 0.885i)6-s + (−0.296 − 0.514i)7-s + 0.955i·8-s + (0.451 − 0.892i)9-s + (−1.11 + 0.642i)10-s + (0.0813 + 0.140i)11-s + (−0.0692 + 0.0426i)12-s − 0.493i·13-s + (0.534 − 0.308i)14-s + (1.08 + 0.588i)15-s − 1.07·16-s + (0.0757 − 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.255 - 0.966i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.255 - 0.966i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.76733 + 1.36142i\)
\(L(\frac12)\) \(\approx\) \(1.76733 + 1.36142i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.55 + 1.57i)T \)
19 \( 1 + (-8.45 - 17.0i)T \)
good2 \( 1 - 2.07iT - 4T^{2} \)
5 \( 1 + (-3.09 - 5.35i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.07 + 3.59i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.895 - 1.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 6.42iT - 169T^{2} \)
17 \( 1 + (-1.28 + 2.22i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 2.45T + 529T^{2} \)
29 \( 1 + (12.8 + 7.40i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (40.1 + 23.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 30.9iT - 1.36e3T^{2} \)
41 \( 1 + (-6.51 + 3.76i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 39.3T + 1.84e3T^{2} \)
47 \( 1 + (-22.4 + 38.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-79.3 + 45.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (66.9 - 38.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-59.4 + 102. i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 51.4iT - 4.48e3T^{2} \)
71 \( 1 + (16.3 + 9.41i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (52.5 - 91.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 46.8iT - 6.24e3T^{2} \)
83 \( 1 + (-1.22 - 2.11i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (47.0 - 27.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 105. iT - 9.40e3T^{2} \)
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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

−13.20311254086188955969502120272, −11.82230110184898758780202195947, −10.51950550810531122938453809768, −9.609634181508503108382907178938, −8.246175592591645464680788662906, −7.33961800624481261084734784488, −6.68963930473474551572846398258, −5.67888590012808085594820273885, −3.51179273973603231067671092535, −2.16350716321523373355233324775, 1.60280083184394013567358958260, 2.85297827912995432186527124553, 4.19930823261328396936005239613, 5.52489543594454466541595368119, 7.22423857551133385028861537536, 8.959296476448034377815871137160, 9.173388932765216417799262796051, 10.23447223147023999661692661041, 11.27629695073101400912551136012, 12.43986094096607477456107743176

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