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Properties

Label 1-693-693.25-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.838 - 0.544i$
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 + (0.669 − 0.743i)5-s + (−0.809 + 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.669 + 0.743i)13-s + (0.309 + 0.951i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.978 + 0.207i)20-s + (−0.5 − 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.913 − 0.406i)26-s + (−0.104 + 0.994i)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 + (0.669 − 0.743i)5-s + (−0.809 + 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.669 + 0.743i)13-s + (0.309 + 0.951i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.978 + 0.207i)20-s + (−0.5 − 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.913 − 0.406i)26-s + (−0.104 + 0.994i)29-s + (0.309 − 0.951i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.838 - 0.544i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.838 - 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4602371883 - 1.553400029i\)
\(L(\frac12)\) \(\approx\) \(0.4602371883 - 1.553400029i\)
\(L(1)\) \(\approx\) \(0.9210599621 - 0.8599996112i\)
\(L(1)\) \(\approx\) \(0.9210599621 - 0.8599996112i\)

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

−23.11879243378584474815837740441, −22.362914756227076302699014426565, −21.46968705059078138581512033310, −21.00276549203141764101005476014, −19.61904652781902580203305854759, −18.557163394726092803647846921605, −18.04562278604806749155196750084, −17.22836038672286471381853201408, −16.46243153590178100007688900897, −15.44444719009395098870516993833, −14.805508504037972636903786880865, −14.01911151561399068537091571777, −13.299002996802840634937218012690, −12.483039013817819308198551942637, −11.322455955237596821724699374498, −10.14971647243575139776028721372, −9.59761934197427663118466335018, −8.20992532610261965525445250738, −7.77066366745214187865467177483, −6.411494639770004510822786328896, −6.04426337136906226883770503128, −5.09295026848283863044015450975, −3.749050544667117934244606775838, −3.06492815036259034927933483814, −1.50652789291116463477407312514, 0.75212391732114212006095292989, 1.823896730263898838463244534140, 2.75837788700215204727244630836, 4.015707012732671534929729056189, 4.8382958071605580113611694220, 5.683776208200368273483272471453, 6.66780005886360404460354066758, 8.21840483264823412322220317465, 9.09968146186841920200925937832, 9.64329792348385102222923189595, 10.66712675643090071648712414549, 11.52983979627994544105913899878, 12.37501877969586459891839655710, 13.14851391902964557920814301070, 13.87206046022955577176252393476, 14.52810798260261329427036477052, 15.82136196602976871740915904079, 16.67770278381631715308042898304, 17.617816375648770469128319431459, 18.41544862739145292167789951852, 19.10928742089286609671308659700, 20.34091007647004835860872348685, 20.52857941467244708889720244614, 21.54589857880654408986422190530, 22.0339779820790495607477087475

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