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Properties

Label 1-693-693.380-r0-0-0
Degree $1$
Conductor $693$
Sign $0.946 + 0.323i$
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  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (0.309 + 0.951i)8-s + (0.5 − 0.866i)10-s + (0.104 + 0.994i)13-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (−0.913 − 0.406i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.978 − 0.207i)26-s + (−0.978 + 0.207i)29-s + (0.913 + 0.406i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (0.309 + 0.951i)8-s + (0.5 − 0.866i)10-s + (0.104 + 0.994i)13-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (−0.913 − 0.406i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.978 − 0.207i)26-s + (−0.978 + 0.207i)29-s + (0.913 + 0.406i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.946 + 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210295136 + 0.2013486963i\)
\(L(\frac12)\) \(\approx\) \(1.210295136 + 0.2013486963i\)
\(L(1)\) \(\approx\) \(1.000262839 - 0.1688597003i\)
\(L(1)\) \(\approx\) \(1.000262839 - 0.1688597003i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−22.66897644117969122484562432302, −21.99628100553958195267513714231, −21.11378974337944429928906893618, −20.23995442158578830096112117204, −19.17445297245042119627253059059, −18.32982179437629627924986843954, −17.38399895239284165395851260228, −17.10302260914501773349741419013, −16.071742532958778075071158882786, −15.30641458707554662868377665861, −14.44536537660967351337448185278, −13.54717289864812500017546128098, −12.96931252681418828530489991199, −12.06593362830324324478410033483, −10.44089906544997301259881316073, −9.93974416415776867351583004617, −8.87542746573690877662930580136, −8.23891631333067773038916769508, −7.26948183137343588551839761643, −6.10232723979842888567443111403, −5.585742982512837272035600997308, −4.660960386120251262753120354566, −3.538704383066019092563627436598, −1.98957965072730215221407781356, −0.64389069758680506941304625708, 1.42394261319058726561537241158, 2.19950158240285227944464778340, 3.25322057358254527582516017358, 4.18936511535657560200225102924, 5.36079015015978807911979236008, 6.27014601856085510538323274692, 7.45300954665064522818939522848, 8.560695860206878865060100038370, 9.49062262266167488756091451304, 10.14122405190455498992791029146, 10.888109436953735616133467255767, 11.87473613975899550523256596921, 12.550581011703914867270653903681, 13.820835307093092995637115128733, 14.00740554301879329305610918293, 15.02148223127010320182857740287, 16.476683404214287379736035703982, 17.14235090818513714307325696242, 18.07586138639784548275358397233, 18.75496711739410393520374723383, 19.2700797252512442550263651637, 20.50567141083681565041826044564, 21.0880184366060762760537597954, 21.74137297075408825629758625373, 22.52607187175419535757299933519

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