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Properties

Label 1-693-693.356-r0-0-0
Degree $1$
Conductor $693$
Sign $0.115 + 0.993i$
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 + (−0.978 + 0.207i)5-s + (0.809 + 0.587i)8-s − 10-s + (−0.669 + 0.743i)13-s + (0.669 + 0.743i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.978 − 0.207i)20-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.809 + 0.587i)26-s + (0.104 + 0.994i)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 + (−0.978 + 0.207i)5-s + (0.809 + 0.587i)8-s − 10-s + (−0.669 + 0.743i)13-s + (0.669 + 0.743i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.978 − 0.207i)20-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.809 + 0.587i)26-s + (0.104 + 0.994i)29-s + (−0.669 + 0.743i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.115 + 0.993i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.115 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.590347038 + 1.416005395i\)
\(L(\frac12)\) \(\approx\) \(1.590347038 + 1.416005395i\)
\(L(1)\) \(\approx\) \(1.527103002 + 0.5445936282i\)
\(L(1)\) \(\approx\) \(1.527103002 + 0.5445936282i\)

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 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
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 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−22.509193921042763927091214195733, −21.89125590104534726789903421920, −20.72677239779085631227236396787, −20.25194563151453189249505819064, −19.381585185354001943819706650791, −18.81393427572718044078671095633, −17.37546087701824278684313769222, −16.56225202464758453291235244959, −15.564299346856250854062422380286, −15.14017110714552163889061425686, −14.27008858195968852405609362283, −13.21010778934566682103778205816, −12.45504525976266366878892324703, −11.87325200084696859602992335912, −10.905800274876195443358737489792, −10.1983094884871898971523633395, −8.874581807836008261345937566019, −7.70856042971193114431959888926, −7.14358744091667793049269918404, −5.88801640133467238594765500684, −5.03224423200051437160871866239, −4.12230089149966282479342928490, −3.30203796168759866930813056163, −2.275225231979766909556231642688, −0.76452238032425099925746081395, 1.52516749088509389456699155284, 2.97153665574600644124370473489, 3.5564437075718845636936245352, 4.71825014175455687420284996352, 5.32499726162106730312393512454, 6.75005717931133360112452708870, 7.26553007249787118805353866228, 8.100314257247640063587752864719, 9.34404816701600818453132249610, 10.544147181813510102197448580552, 11.58795889098553962104986998959, 11.921054213384163445084896858156, 12.85727299426485084490746566952, 13.96045040339509806828362785743, 14.53417512985380973652453477012, 15.372758702331945843828308512418, 16.22166635285361047836205304818, 16.67614083818620182341914173236, 17.98240612252528048444109312617, 18.98165153532816176533223176492, 19.80660465578338306134334932929, 20.44227723246802928339008929176, 21.42074182773503273279434393565, 22.16967144495022912351669332380, 22.956025886466825841843669594668

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