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Properties

Label 1-693-693.283-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.152 - 0.988i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.152 - 0.988i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.152 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547891572 - 1.804481886i\)
\(L(\frac12)\) \(\approx\) \(1.547891572 - 1.804481886i\)
\(L(1)\) \(\approx\) \(1.594054766 - 0.7522049403i\)
\(L(1)\) \(\approx\) \(1.594054766 - 0.7522049403i\)

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

−22.84800132795184834088940435418, −22.251537020543468461087653124662, −21.43514263364700237376699608285, −20.71803909147991661828581277556, −19.576751107416053844305641795651, −19.10206400084155241744770633437, −17.934963324657026676064143318140, −16.94803043297351395882985351737, −16.24530549812161194555763757091, −15.03769139836511078651288122903, −14.78731611336676601403705505217, −14.01930654127644309948483254761, −12.83458166375642724529215816444, −12.27613985441691184787391203273, −11.23579003756696454162397374900, −10.60720498241078842451152479487, −9.590753942536407216611938845676, −7.95713328658588371917292163310, −7.51158688754498602189220186075, −6.47856186806685828179713154402, −5.71103665365870415397457369965, −4.62631268611984144270663378689, −3.591518170217294438317501660738, −2.88780130808336526963002414543, −1.76345300685500697720712851130, 0.81686369999560757331786320023, 2.117432662222391978941778409524, 3.159268309334743583552231850701, 4.30404603424474896218714610614, 4.98912885668614671905238722872, 5.72916237316519667145122709289, 7.08574452575894774287267454478, 7.66197030832443180527951287915, 9.08963925451947467850460047783, 9.76629758544463119231464922616, 11.12216542951319609623551144511, 11.65392185537559527089408027431, 12.713320941990163856813950786796, 13.03083714151492083765083008857, 14.18287717917786737762653756051, 14.922555449415257804211545903704, 15.783087269209314522640668507863, 16.59101156222879194111761650255, 17.20963551200702760130232547786, 18.63913495183862121152051238748, 19.5158498624786305020594228814, 20.15421444988383711992217294761, 20.87244318978128938354851536822, 21.63759631420473239513123282193, 22.43071821989021306047943365493

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