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Properties

Label 1-693-693.520-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.941 + 0.337i$
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  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)5-s + (−0.809 + 0.587i)8-s + (−0.5 + 0.866i)10-s + (0.669 + 0.743i)13-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.978 + 0.207i)20-s + 23-s + (−0.809 + 0.587i)25-s + (−0.104 + 0.994i)26-s + (−0.104 + 0.994i)29-s + (−0.978 + 0.207i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)5-s + (−0.809 + 0.587i)8-s + (−0.5 + 0.866i)10-s + (0.669 + 0.743i)13-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.978 + 0.207i)20-s + 23-s + (−0.809 + 0.587i)25-s + (−0.104 + 0.994i)26-s + (−0.104 + 0.994i)29-s + (−0.978 + 0.207i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.941 + 0.337i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (520, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.941 + 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3260019442 + 1.874569331i\)
\(L(\frac12)\) \(\approx\) \(0.3260019442 + 1.874569331i\)
\(L(1)\) \(\approx\) \(1.036863930 + 1.039904026i\)
\(L(1)\) \(\approx\) \(1.036863930 + 1.039904026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
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

−22.26979901679879115297353323999, −21.36901501576351835151421323508, −20.73447943676572269956006762939, −20.04156788425898998597101920975, −19.42011236710040833791375047810, −18.20018844531532976888243578857, −17.6393555625347737355912465989, −16.42191501113402742961729841849, −15.60847960116547687735696888931, −14.82550902594377899724925553068, −13.61531306658269234329430885284, −13.19007909562393159764567039239, −12.49549238487525442877314845535, −11.42463596639620856809526086753, −10.80525610057083178187668615099, −9.614550490996291814570170712455, −9.07190066424556025084624094594, −7.9668014975367334929776658665, −6.55113538793600544747077757067, −5.62400513516387332070682334373, −4.90182139371322971445002398879, −3.969161431902899631821673120817, −2.89471680731379776948140442537, −1.7580178602627307608759356427, −0.73236015052467357623142897709, 1.86484045708542058493386809792, 3.06162553178096620554083053560, 3.7943544168829110066602227531, 4.994501983961960082384854759543, 5.8822478584330981171506898999, 6.8793720753566035373403216615, 7.23017232880382790374311572857, 8.5908525981017493674801757577, 9.31565136094997125064875875542, 10.658836864649026131578989105850, 11.40047863117902031870803973975, 12.3162222381332935919945435984, 13.56501314518852854748430366185, 13.80376296087827875125115273724, 14.852237180971930086768050131238, 15.44956821129203472284722479645, 16.396699763790931913508700943858, 17.15148032559897673735202522140, 18.259283014790774814327945376570, 18.51340692891888107110891883889, 19.93257319274126326269570397817, 20.8629800790351217914833933018, 21.69386146080305246653955938545, 22.2964526174339012559495758680, 23.0035063647127197874829959300

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