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Properties

Label 2-171-171.121-c1-0-11
Degree $2$
Conductor $171$
Sign $0.988 + 0.152i$
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  + (0.5 − 0.866i)2-s + (1.5 + 0.866i)3-s + (0.500 + 0.866i)4-s − 5-s + (1.5 − 0.866i)6-s + (−1.5 − 2.59i)7-s + 3·8-s + (1.5 + 2.59i)9-s + (−0.5 + 0.866i)10-s + (−1.5 − 2.59i)11-s + 1.73i·12-s + (3 + 5.19i)13-s − 3·14-s + (−1.5 − 0.866i)15-s + (0.500 − 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 + 0.499i)3-s + (0.250 + 0.433i)4-s − 0.447·5-s + (0.612 − 0.353i)6-s + (−0.566 − 0.981i)7-s + 1.06·8-s + (0.5 + 0.866i)9-s + (−0.158 + 0.273i)10-s + (−0.452 − 0.783i)11-s + 0.499i·12-s + (0.832 + 1.44i)13-s − 0.801·14-s + (−0.387 − 0.223i)15-s + (0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.988 + 0.152i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69941 - 0.130499i\)
\(L(\frac12)\) \(\approx\) \(1.69941 - 0.130499i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
19 \( 1 + (4 - 1.73i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.5 - 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.97521832623334953220265396399, −11.59094825265085010171728403008, −10.86324642580606885432002952431, −9.920853486335495210077965541604, −8.598884274685412542893589621993, −7.74510585684968636519446495978, −6.56471602709823055667951663543, −4.20616057459367871776702764288, −3.86507867123677708209267961929, −2.38020460520154617673660805410, 2.11777052431038888001191696307, 3.72366236420306153594793405401, 5.48619817971487371812528013787, 6.43824348244610995544157926519, 7.60034681006466857306184100899, 8.402380373340848535792101374326, 9.662682133372304131530054979206, 10.70703578334391991843956196728, 12.14026648587108730965363885108, 13.01135616862345496556186435003

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