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Properties

Label 1-693-693.268-r0-0-0
Degree $1$
Conductor $693$
Sign $0.601 - 0.798i$
Analytic cond. $3.21827$
Root an. cond. $3.21827$
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  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)5-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)10-s + (−0.978 − 0.207i)13-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.669 − 0.743i)20-s + 23-s + (−0.809 − 0.587i)25-s + (0.913 + 0.406i)26-s + (0.913 + 0.406i)29-s + (0.669 − 0.743i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)5-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)10-s + (−0.978 − 0.207i)13-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.669 − 0.743i)20-s + 23-s + (−0.809 − 0.587i)25-s + (0.913 + 0.406i)26-s + (0.913 + 0.406i)29-s + (0.669 − 0.743i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.601 - 0.798i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.601 - 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8430089139 - 0.4203704958i\)
\(L(\frac12)\) \(\approx\) \(0.8430089139 - 0.4203704958i\)
\(L(1)\) \(\approx\) \(0.7462421037 - 0.1870158411i\)
\(L(1)\) \(\approx\) \(0.7462421037 - 0.1870158411i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−22.93711721440484646282717553875, −21.78632036794476607418817042243, −21.21199421540807928394265938720, −20.11255499869547642235601075789, −19.25858637441670454106589387854, −18.805674813022813602110029692362, −17.72083390495823635865783183011, −17.39558115259848308517016208266, −16.302652491421814500904133136412, −15.4789014450923194262374239125, −14.628509843522330503433811276699, −14.04207988173111246964972154398, −12.69422335524617436801112346089, −11.56221376817056384297784839313, −11.0129148618071833148939150472, −9.92033653560973962424373145322, −9.53297923573777520881580521309, −8.361052041301500976073634670907, −7.286589938893483692468230451946, −6.85967273731711497431179573452, −5.81826553772847047378614031126, −4.72068967556776558418730498077, −2.9243062014871916737913239850, −2.51323980240321290993172155149, −1.00831624486164999106598298831, 0.79958852859226832634274662098, 1.81059651119284476471628608551, 2.89581726770490519350739085583, 4.17588662249813029770090557221, 5.38282407634624658811023930624, 6.290149764747488350703719485972, 7.49643318923943195965903553145, 8.21615381990637653941355749724, 9.037339480783743579255582499110, 9.91364735146065172087452710158, 10.48457862973114274860038097734, 11.78893748038242361435499895161, 12.38498372518700264228528890162, 13.12183574808013783440386840428, 14.446936399693913970763739849648, 15.32175619442919712264315630834, 16.328627444971900215458085557021, 16.99959913637363354070780021952, 17.43191918635935314671495760171, 18.539334545053405423994057175812, 19.32847727859266271766737406254, 20.00404803671999819057753277819, 20.91081738554749932670981840210, 21.33711392038380927858912568024, 22.38294222778772933785176695744

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