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Properties

Label 1-693-693.38-r0-0-0
Degree $1$
Conductor $693$
Sign $0.999 + 0.0111i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.104 + 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.104 + 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.999 + 0.0111i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.999 + 0.0111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.213881660 + 0.01237308261i\)
\(L(\frac12)\) \(\approx\) \(2.213881660 + 0.01237308261i\)
\(L(1)\) \(\approx\) \(1.618373319 - 0.2038402562i\)
\(L(1)\) \(\approx\) \(1.618373319 - 0.2038402562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
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

−22.908857514787956245096668635411, −21.942015187597164814853685290661, −21.10484546932752974678234301799, −20.31755155391102220399966061829, −19.87573486247376755282297904132, −18.32589451253294603362136137441, −17.587315844773553370765030557957, −16.71185886426569663524082414656, −15.966349298681529609133972757072, −15.448194954698315154074779993047, −14.3259079919386297692609343951, −13.54105014842531231162724828352, −12.80844814299202206428276372516, −12.08394628937959625017773344564, −11.28516253957277109146090261520, −9.97471066384843855063814932736, −8.84358689083618125626402938734, −8.145950259343070266876452602468, −7.25424955174794559743089202334, −6.22766763084551298579239174760, −5.10722852761659233975230206421, −4.78736901260554166463130038158, −3.48447728329455576560395242052, −2.57781176463634300448845659211, −0.90542689687867168897558352780, 1.373332346507290039085328353473, 2.43570526421544571971300286684, 3.39423063612475982125602097569, 4.16114376899510061188800282013, 5.30708956733430886676836588758, 6.35678011852847890792407422291, 6.92953896198014103717194473571, 8.16360835011232837878126546767, 9.58280423426148386923772287512, 10.162022126837895394322211623265, 11.263225296393655920906254398737, 11.62384992652217282254950285417, 12.679512876261690756247570033928, 13.786538874296533483579858781712, 14.13346127876365704661949371374, 15.21348941595111058874368549963, 15.670751439703252286965810495437, 16.93885217723983471926630639389, 18.016899945518633732483544549205, 18.96052249160934014101249630173, 19.30031222209300461458770950223, 20.34485206112707674288525755824, 21.24496983580773063803100513419, 21.92339965216829295883727346650, 22.49129574233981375857861659314

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