Properties

Label 1-165-165.98-r1-0-0
Degree $1$
Conductor $165$
Sign $-0.525 - 0.850i$
Analytic cond. $17.7317$
Root an. cond. $17.7317$
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  i·2-s − 4-s + i·7-s + i·8-s i·13-s + 14-s + 16-s i·17-s + 19-s i·23-s − 26-s i·28-s − 29-s + 31-s i·32-s + ⋯
L(s)  = 1  i·2-s − 4-s + i·7-s + i·8-s i·13-s + 14-s + 16-s i·17-s + 19-s i·23-s − 26-s i·28-s − 29-s + 31-s i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(17.7317\)
Root analytic conductor: \(17.7317\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 165,\ (1:\ ),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6705897548 - 1.202772797i\)
\(L(\frac12)\) \(\approx\) \(0.6705897548 - 1.202772797i\)
\(L(1)\) \(\approx\) \(0.8321837114 - 0.5143178185i\)
\(L(1)\) \(\approx\) \(0.8321837114 - 0.5143178185i\)

Euler product

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

−27.54781708491607591875236856917, −26.37215080772380649567862236725, −26.15881490594634339795885434275, −24.75982396071693124747809454550, −23.90822206762110823078034665385, −23.22690441476933078785630445133, −22.16457182369597443521470204450, −21.12809871213120954911274905348, −19.76919298340128410964825669840, −18.81157149507894008161776265835, −17.60446945950856150624461983548, −16.86076526781270705460777727938, −15.9816523597483021045968647789, −14.82496082144680879078619938485, −13.86399256773000493048359533350, −13.12744183543687765270139142570, −11.62633049286044270119819786381, −10.20239604886977899424287647472, −9.23447318967162562149966839126, −7.92805920216643302180845828219, −7.066471377864752798098239591864, −5.96679628099446702631375798568, −4.598654041246287136158843543898, −3.6068061043085927690427802255, −1.22783501663600599305599946791, 0.592390944305486126694877326797, 2.29825558779942820920051891176, 3.26257337971883407658062806349, 4.84945140854833084053675748173, 5.805642731287421035257223493865, 7.706412750330417359887615598000, 8.88245697370872160345176696680, 9.76337474385523707305302519383, 10.95997687054846472237890357861, 11.97086960178986109703733464891, 12.76172107331072098161633993374, 13.90885902981808570530502081238, 15.00475423474196542521525718521, 16.18829089147065899764157769842, 17.710529267769294214278100874209, 18.33720185425231069276553433890, 19.293686931729587021327684333320, 20.41520468890933644356747381730, 21.08670863592837037053099107488, 22.479667106301085516886261663882, 22.58806011853645320977265251252, 24.24858369160853934172609091885, 25.15319393382140663413374661314, 26.42554816277733087976062647225, 27.35803708671553549803793558547

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