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Properties

Label 2-171-171.14-c1-0-13
Degree $2$
Conductor $171$
Sign $0.718 + 0.695i$
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  + (1.12 + 0.942i)2-s + (−1.13 − 1.30i)3-s + (0.0260 + 0.147i)4-s + (−0.668 − 1.83i)5-s + (−0.0404 − 2.53i)6-s + (1.13 − 1.96i)7-s + (1.35 − 2.34i)8-s + (−0.426 + 2.96i)9-s + (0.980 − 2.69i)10-s + 2.71i·11-s + (0.163 − 0.201i)12-s + (−0.159 + 0.437i)13-s + (3.12 − 1.13i)14-s + (−1.64 + 2.95i)15-s + (4.01 − 1.46i)16-s + (0.305 + 0.838i)17-s + ⋯
L(s)  = 1  + (0.794 + 0.666i)2-s + (−0.654 − 0.755i)3-s + (0.0130 + 0.0737i)4-s + (−0.299 − 0.821i)5-s + (−0.0165 − 1.03i)6-s + (0.429 − 0.743i)7-s + (0.479 − 0.830i)8-s + (−0.142 + 0.989i)9-s + (0.310 − 0.851i)10-s + 0.817i·11-s + (0.0472 − 0.0581i)12-s + (−0.0441 + 0.121i)13-s + (0.836 − 0.304i)14-s + (−0.425 + 0.764i)15-s + (1.00 − 0.365i)16-s + (0.0740 + 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26627 - 0.512359i\)
\(L(\frac12)\) \(\approx\) \(1.26627 - 0.512359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.13 + 1.30i)T \)
19 \( 1 + (-4.01 - 1.68i)T \)
good2 \( 1 + (-1.12 - 0.942i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (0.668 + 1.83i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.13 + 1.96i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.71iT - 11T^{2} \)
13 \( 1 + (0.159 - 0.437i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.305 - 0.838i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (6.24 - 1.10i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.09 - 6.19i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + 5.62iT - 31T^{2} \)
37 \( 1 - 6.58iT - 37T^{2} \)
41 \( 1 + (-6.32 - 5.30i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.22 + 6.93i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.78 + 0.667i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (9.68 - 8.12i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-0.779 + 4.42i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (8.05 + 2.93i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.97 - 2.35i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.0740 - 0.0621i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.14 + 6.48i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-4.08 - 11.2i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (3.19 + 1.84i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.41 - 8.02i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.63 - 3.13i)T + (-16.8 - 95.5i)T^{2} \)
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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

−12.64582225579678419859568375358, −12.10772495848609528259849602599, −10.81296769313728318082788075180, −9.726710245524060406412173189008, −7.979087776483739487975268622360, −7.29828434826092866550122583405, −6.14674662532033373421393947236, −5.05072332290302342519774677656, −4.24648551804306007702727436657, −1.30120968495450581088796209069, 2.77656952526793572910944849757, 3.84476445356548423452198456959, 5.10070065445881250755055289198, 6.07218611395480255944461720743, 7.71421892405444624833685180859, 8.984672270890887049053414712484, 10.35133784369572930301510625379, 11.22623859827155958165055862751, 11.70489220441105394182475713111, 12.55377451144934469572142026393

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