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Properties

Label 1-693-693.58-r0-0-0
Degree $1$
Conductor $693$
Sign $0.999 + 0.00469i$
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  + (0.309 − 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

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

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.999 + 0.00469i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.999 + 0.00469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7742958570 + 0.001817544670i\)
\(L(\frac12)\) \(\approx\) \(0.7742958570 + 0.001817544670i\)
\(L(1)\) \(\approx\) \(0.7560734243 - 0.3237569177i\)
\(L(1)\) \(\approx\) \(0.7560734243 - 0.3237569177i\)

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

−22.71786538954404034693309742844, −22.19088078525116115551165710310, −21.33147179707650442812048343039, −20.01227978080699001451076216537, −19.522955273911546033457354201073, −18.31893323481235089679151945770, −17.766681835869337789199593543545, −16.74862879123501278445846134807, −15.91194728411872015580244490150, −15.40119010765576054548458758671, −14.50438143221079895269333463615, −13.84328842766325697454359979116, −12.609786593488866609762916608676, −12.14660586447573468539134043982, −11.03223199537166157827424084398, −9.92135144337091823345268067410, −8.79921532928017322948369373253, −8.11354333277037494681443769525, −7.14461712256798267688432933393, −6.64320989022292010557207109455, −5.270623860315074053678835363275, −4.55191210998233018149353875859, −3.60904622365302516495353053129, −2.56895552236717434269055349159, −0.408129479863735092741632375397, 1.04966757226022147512811794096, 2.38554274393182358908943419698, 3.33136071809010407497696242654, 4.36001568253809362477186873602, 4.9384520226036346070810390678, 6.19839632685428619730775349169, 7.48184958871861235333424844005, 8.317151868937451202004532680593, 9.40576913305278391832088941312, 10.05750181213831611580150169489, 11.33296075941238080409204542914, 11.69133733947640122318679100386, 12.52070516047650677722112094463, 13.4184637235296723403238003989, 14.285317331902297322663952302318, 15.1931315498307463884805336916, 15.91855996630751102527150208382, 17.08012957663225032631569862452, 17.98940333470514174789032539532, 18.84280356032617380883165556621, 19.724971585354678713511220333381, 20.01010140220644447359273962551, 20.994949819275007526688101487763, 21.89355270146493442923073591396, 22.60472774476545653328321483816

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