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Properties

Label 2-171-171.151-c0-0-1
Degree $2$
Conductor $171$
Sign $-0.642 + 0.766i$
Analytic cond. $0.0853401$
Root an. cond. $0.292130$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s − i·3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + i·8-s − 9-s + 0.999i·10-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 − 0.499i)14-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)18-s − i·19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s − i·3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + i·8-s − 9-s + 0.999i·10-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 − 0.499i)14-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(0.0853401\)
Root analytic conductor: \(0.292130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3799375747\)
\(L(\frac12)\) \(\approx\) \(0.3799375747\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
19 \( 1 + iT \)
good2 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−12.50378611910378692433937388903, −11.59210384501217424320819093772, −10.87874465828559044037538146960, −9.109533802221884348670597627972, −8.837033921878308270395272134693, −7.88754360637046314271475907853, −6.29119669582383274707931373042, −5.25063307880484960493204905994, −2.99343008038472063889121925162, −1.13558389195596783823694363533, 3.50118570811506833453859075622, 4.23496781639815852337266858220, 6.37343530522119538900024165060, 7.20151037666662089677640548508, 8.336196475863634597614208489047, 9.401579418067524432331958793208, 10.20140067938159415834233222041, 10.95234448530257216624214282251, 12.18579858521529962424400026850, 13.53765669200239560920799691729

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