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Properties

Label 1-693-693.367-r1-0-0
Degree $1$
Conductor $693$
Sign $0.758 + 0.651i$
Analytic cond. $74.4731$
Root an. cond. $74.4731$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.978 + 0.207i)13-s + (0.309 − 0.951i)16-s + (0.978 − 0.207i)17-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.104 + 0.994i)26-s + (0.913 + 0.406i)29-s + (−0.309 − 0.951i)31-s + 32-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.978 + 0.207i)13-s + (0.309 − 0.951i)16-s + (0.978 − 0.207i)17-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.104 + 0.994i)26-s + (0.913 + 0.406i)29-s + (−0.309 − 0.951i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(74.4731\)
Root analytic conductor: \(74.4731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (1:\ ),\ 0.758 + 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.647729067 + 0.9800659207i\)
\(L(\frac12)\) \(\approx\) \(2.647729067 + 0.9800659207i\)
\(L(1)\) \(\approx\) \(1.334937212 + 0.5639099594i\)
\(L(1)\) \(\approx\) \(1.334937212 + 0.5639099594i\)

Euler product

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

−22.04039510037612925610246208868, −21.55538062804224888114662277182, −20.90697776900145259978762159100, −20.088237789633366573212038905577, −19.12901034438570451469583280199, −18.41417197449958583692808276776, −17.68312535028254049177367520012, −16.91723330498582828379987274298, −15.60899641662050974682149584741, −14.68697472533284021880336236639, −13.82248004118884667867185964649, −13.32232427087317948141201932704, −12.4076918579399594971393917648, −11.49563983435750778422194749485, −10.48336148137959934485126609184, −10.06677411787545066420634661022, −9.01470398695150165345704795496, −8.234008439089132777789929367077, −6.6421709364802265419029198681, −5.81853886279077279625331095311, −5.021656034949655206464358140188, −3.79254528317565054754319989948, −2.926384920775196442622041013074, −1.8491516038531211668326792275, −0.9933688035214323105365494811, 0.70885403910478471791383439893, 2.11371692983017508510921620168, 3.45635513008488608000740339055, 4.44836032228134808395616059152, 5.50879420604035225836239524127, 6.13616471712851348909019139089, 6.94090174652337558036055728671, 8.1892967990504261946049180714, 8.78721651460016447267275461576, 9.778057425486343903808144139055, 10.609098756283150988391773371804, 12.04870227631058172939694991982, 12.81392706716919068418147583855, 13.63594571178407498672258166697, 14.27421692165334931386460794817, 15.03859805947609692031958017768, 16.23003162111020013572450126831, 16.62793702120006656659375054770, 17.53244301173236362778267480610, 18.31587410490430778563213364019, 18.933689580243507959384067270, 20.46934135586460068666501962809, 21.12079809908748784180332458472, 21.7993287954622472547409214529, 22.67952522537950082598894583660

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