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Properties

Label 1-693-693.16-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.998 + 0.0521i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.998 + 0.0521i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.998 + 0.0521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01967132764 - 0.7535543679i\)
\(L(\frac12)\) \(\approx\) \(0.01967132764 - 0.7535543679i\)
\(L(1)\) \(\approx\) \(0.5649721267 - 0.5134491881i\)
\(L(1)\) \(\approx\) \(0.5649721267 - 0.5134491881i\)

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

−23.127126974852128291073213177308, −22.630702920744156940179862466045, −21.67758738986412424718104689661, −20.72866124145900319787841971076, −19.40609485831321425508173527186, −18.90683908391638798108473208409, −18.30096579522889430883529311156, −17.11504601973371851130305985470, −16.51701580585773306157942790463, −15.69098665570896569740258936795, −14.844664428917284804294960534435, −14.331670251291558668571261543561, −13.3825715584377269175764782300, −12.2728307044055963284578611760, −11.45807498925488779638733414397, −10.315770466496171190501922854150, −9.49914752676790085351565017008, −8.4393356504794915015969200521, −7.66508057311763402684269982725, −6.94963030611560359992717477090, −6.07713281565842099514740282421, −4.98725132869554409909798207160, −4.00219414661707344709810106094, −3.1652578603528413735547644607, −1.36703674585220763885427709935, 0.434604954690200542786657399656, 1.48679970993998824385731392375, 3.03427624835847284887833988453, 3.53718383605014751062070082267, 4.9188824990493018552470531746, 5.285921384127766497897194899502, 7.12039209964190555667550386390, 8.010230214677808094339837670, 8.77161640802866165417464837284, 9.67568091666671344307644916095, 10.580243477372591566589105084446, 11.50946258850727598366827075926, 12.129124537635395246506567380543, 12.94130660052036999464349984985, 13.662535863916968140878315038037, 14.826745195166632599706064173484, 15.64155716132731685169508381579, 16.71610520223807112695641110161, 17.42371410568244094919363648278, 18.420230411987238754033850671574, 19.1515218408274108310368552534, 19.912487680301581312966647048566, 20.56422948983345313215344939676, 21.171811649191017598689524806951, 22.35283086641730000974957341405

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