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Properties

Label 1-693-693.409-r0-0-0
Degree $1$
Conductor $693$
Sign $0.374 - 0.927i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.374 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.347188961 - 0.9086463780i\)
\(L(\frac12)\) \(\approx\) \(1.347188961 - 0.9086463780i\)
\(L(1)\) \(\approx\) \(1.078672105 - 0.5745297660i\)
\(L(1)\) \(\approx\) \(1.078672105 - 0.5745297660i\)

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

−22.721197799729380822650979686129, −22.324612516949699788762342386973, −21.405388539790073538548685820788, −20.596122179408186716181058036522, −19.2932008762422081386980254592, −18.49339435280815508333327533750, −17.74209406664359494746408297681, −17.21222555502585815105546201715, −16.22981045835035307129245321119, −15.335628197353094031271082227106, −14.64732534267212362522522524599, −13.83865080582836809962232902584, −13.18223600635462289456730832503, −12.25214708581834203035940640744, −10.93269144885748271823556882546, −9.97107029729149349518842416340, −9.33034732374657927030665622365, −8.24745212430844286729481988178, −7.28401007921682501394780125237, −6.63095508149299192560295118478, −5.4895442545444536569199432744, −5.096003931578532783017983355332, −3.53630410242830900541601395129, −2.71890269077378608602210636175, −0.984014697034880578629260579345, 1.14042353451572434427668506626, 1.87845477939530723942372537246, 3.06004296937294299182857687671, 4.103852655107140071068102273734, 5.139549192664458185126657530933, 5.783545326249250888711048752374, 7.13170374137212816299040037742, 8.51975952468572790092423349588, 9.103646196545145456195256178322, 10.00790945398050828209548330902, 10.63448328465411256304699671353, 11.90612942369799520738556594744, 12.35318393573631269830480630981, 13.34349137857540577488457793631, 14.05164227124335881925547953506, 14.69496945124897843854229088469, 16.17836467050704757854253060506, 16.9700803448812490713457404263, 17.683588296585274296145618534909, 18.6719007913998438277084684100, 19.22254150504255914335829799532, 20.33785250125437138250101400249, 20.87709722751115734083387311015, 21.58779895778735448976696569146, 22.20024115596340378720071858269

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