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Properties

Label 1-693-693.247-r0-0-0
Degree $1$
Conductor $693$
Sign $0.956 + 0.292i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.104 − 0.994i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.978 + 0.207i)26-s + (0.669 + 0.743i)29-s + (−0.809 + 0.587i)31-s + 32-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.104 − 0.994i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.978 + 0.207i)26-s + (0.669 + 0.743i)29-s + (−0.809 + 0.587i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.956 + 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.234832372 + 0.1846493725i\)
\(L(\frac12)\) \(\approx\) \(1.234832372 + 0.1846493725i\)
\(L(1)\) \(\approx\) \(0.9391877182 + 0.1362067130i\)
\(L(1)\) \(\approx\) \(0.9391877182 + 0.1362067130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.104 - 0.994i)T \)
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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

−22.4314219274568110359148871470, −21.63062850541050702846882719902, −20.86373408281936450193156020919, −20.35916675677884284447289270030, −19.214269861828523833653767996, −18.40194130748712610761121158873, −18.039447259037198895698083495241, −16.99944984566521273783527927744, −16.40909879047595825192353585807, −15.344811778183546247296900187951, −14.211738288353240519757339968550, −13.43243302853912987564986854917, −12.517314003929701148143937171070, −11.68261263641027278554029530778, −10.517746126622074906043064857736, −10.23018480049521207358748154708, −9.23605642982585450690558000831, −8.33602215503313889458685626895, −7.47117688007378254762637128828, −6.35158394977752496674412421654, −5.574398077086400700870791170874, −3.96033451161504891869368450636, −3.081931920664073234295275724805, −2.0452626592750207657852824873, −1.07836273639080953393265490302, 1.05251023743622089278350799466, 1.859270010562632391162806078, 3.24585345028357553416291816150, 4.89115005888106787900337581676, 5.57693445225362137803234192705, 6.4637885683236604731056654513, 7.35334568398273824778363673382, 8.421635367679705740523489204093, 9.19602120994746411102009548209, 9.819359034640935578788918093480, 10.75620041099106405037414686230, 11.66605381752157360975647001528, 12.85716315012065025296272086049, 14.005236176879542910010414775, 14.23226045155384905618198177800, 15.72144993346135242646685720913, 16.1357945040776060254280645311, 17.045629626884779812176756661880, 17.846420534424823948016254682283, 18.33925260013839427055086208923, 19.33895630481086537919995849814, 20.22057178603377009299729894030, 20.95897413067832560032479113516, 21.78877785389598469596796275859, 22.91889892148613305144189738823

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