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Properties

Label 1-693-693.212-r1-0-0
Degree $1$
Conductor $693$
Sign $0.00469 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00469 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00469 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.00469 + 0.999i$
Analytic conductor: \(74.4731\)
Root analytic conductor: \(74.4731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (1:\ ),\ 0.00469 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.391404106 + 1.397951705i\)
\(L(\frac12)\) \(\approx\) \(1.391404106 + 1.397951705i\)
\(L(1)\) \(\approx\) \(0.9514991692 + 0.4969040721i\)
\(L(1)\) \(\approx\) \(0.9514991692 + 0.4969040721i\)

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

−21.97556070640701472307780747684, −21.46219994455171227968670379958, −20.60313717502201659994535673871, −19.96887309806067371272546839273, −18.99645947020321365077809748889, −18.289396995806529481944322856377, −17.30792821062147928813953423766, −17.04358574202146331013850899565, −15.825622380393254620740904204402, −14.46905443765129631915189204361, −13.859537288012681686483720380625, −12.97716808773556881560021877931, −12.25877051721814054697359944880, −11.38242044184213812279152380450, −10.340252356523956125996406651751, −9.65622591684299380670830474366, −9.11278369405642232503977332515, −7.91235729829268037629318702116, −7.033210708896354726932783033210, −5.44884817629181297900939622, −5.01950438200562613826390146472, −3.567266904574313461539355742024, −2.67146772118413858136451521274, −1.682276612583694398844999862808, −0.66784327965445943656246905870, 0.88288758798442281881666896302, 2.052415649086815575239371026296, 3.41554635054488888656471561938, 4.849731634903253580291481007820, 5.483272800661029789627235462191, 6.416995806889980459785547481359, 7.23491971167263030597053427390, 8.11932600743509143872151861946, 9.23032689841319933212466977655, 9.85994783685776441159758844374, 10.51662143092917972242000751602, 11.913705569672321251195724385900, 12.99705091176316831288830020299, 13.76370082520327170774871367276, 14.61652755702344107796928984979, 15.037347555181500199159583441652, 16.45028147856412530975444497054, 16.82813883148626161981221333429, 17.63240635681077190825963345844, 18.551492145316678634276427943350, 18.987478427240905167167275074230, 20.24351161525675189404139791323, 21.12476433065867941291900392730, 22.266069431490843231453996355615, 22.481283756833919031654050530317

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