Properties

Label 1-171-171.140-r1-0-0
Degree $1$
Conductor $171$
Sign $0.466 - 0.884i$
Analytic cond. $18.3765$
Root an. cond. $18.3765$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + 25-s − 26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + 25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.466 - 0.884i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ 0.466 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3203589281 - 0.1932360528i\)
\(L(\frac12)\) \(\approx\) \(0.3203589281 - 0.1932360528i\)
\(L(1)\) \(\approx\) \(0.7361433900 + 0.3881967632i\)
\(L(1)\) \(\approx\) \(0.7361433900 + 0.3881967632i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−27.56118971466880401205392079811, −26.88602849960409284354622970758, −25.57736429398117384444398879769, −24.20893297788600976954605758987, −23.22322791687624947806386540838, −22.82604415423925192160762233808, −21.724901707872395811741914387559, −20.354405702552995325173188951217, −19.84936815528764017311097353728, −19.14269680092542602981021532353, −17.80221337029518284792602965327, −16.64086870051526409183986274271, −15.212223310188854430514490406347, −14.62725670514531007852652561638, −13.14239392191189582562774785049, −12.49476506737563019004686122692, −11.41381096853261583104154309409, −10.40022156162828615811219307036, −9.48660919817983729158826711823, −7.91278854222780482175650711995, −6.75722076489534543449464459450, −5.14335923189624539581658529235, −3.97600379712005554694209960253, −3.20756771153680617365495951047, −1.34840593407694435687662371414, 0.12217267335936791835695664579, 2.85765861608365288988947984172, 3.94120719830756765645403460831, 5.17568255856431074075757285702, 6.392559107359956999873264292575, 7.383429470073493973507686818718, 8.57445900458429015629289314670, 9.38167798837874390940871718707, 11.465050429865158283332482087965, 12.08280505722028991733909865013, 13.20693959285195956154254772449, 14.479346100411251324486031597914, 15.18763539074593398274003465929, 16.35991089677683437519185636418, 16.70229363644683525929224212714, 18.52994123088347773399656688578, 18.98333296426916159840774126340, 20.432767662209427942477341791431, 21.743750190119719614164147368212, 22.376355956814798703019343732780, 23.353265520558888482486362801011, 24.315766678186850930729014793961, 24.95475927881933030540876397769, 26.156365311551262805544707238729, 26.966739460917493809996019960381

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