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Properties

Label 1-693-693.389-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: $\chi_{693} (389, \cdot )$
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

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

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