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Properties

Label 1-693-693.563-r1-0-0
Degree $1$
Conductor $693$
Sign $0.927 + 0.374i$
Analytic cond. $74.4731$
Root an. cond. $74.4731$
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

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

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(74.4731\)
Root analytic conductor: \(74.4731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (1:\ ),\ 0.927 + 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6194734808 + 0.1204278945i\)
\(L(\frac12)\) \(\approx\) \(0.6194734808 + 0.1204278945i\)
\(L(1)\) \(\approx\) \(0.5648890763 + 0.3595325358i\)
\(L(1)\) \(\approx\) \(0.5648890763 + 0.3595325358i\)

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

−22.39178161202903480487029474411, −21.5713387280558948666720945420, −20.50121519579315409581603603143, −20.01009587642773277181007708811, −19.49985670856876096488544434169, −18.39825451566595407106700663019, −17.72580013681063335844518004718, −16.842371526476374156377461273715, −15.8192068738777668580697750796, −15.03080967251451753117193861168, −13.90289524606391750559312636670, −12.917346851776724875328855124564, −12.51717989965578993545387840849, −11.40811703645581857586963791420, −10.94187879394615247605168855905, −9.783106636803038363897747170, −8.9731513294663778007745819012, −8.13204138412962034983862850661, −7.37039545756577733990286795150, −5.73437842692206677505443643168, −4.803235776200112595542704047541, −3.94236738675638819443986213457, −3.06507672400665985787057273233, −1.84344292731751150479591124462, −0.63709967114583135469204994203, 0.25223876837190570275740490872, 1.9437500824212572848281594021, 3.559515717498308237716549575584, 4.20906691386139500665752129136, 5.31972835536866836622983998059, 6.40178956487103294022148633375, 7.15968846268368004031669005024, 7.83340579951311592781842185721, 8.83925346435263836162895824094, 9.65301867562708336132482237956, 10.72975125033870469244860536850, 11.67553624274738657802060713879, 12.584334642178556163312831083260, 13.80120291189280919788986448109, 14.3362931831381459859336008533, 15.177173762705939079826959058804, 16.05430240261731167363866711951, 16.45304042862562713365345063920, 17.69366267636794959437445788845, 18.327678681335776752181025212906, 19.091660291153192786411274264, 19.81388683226739065631726320186, 20.96150942518565460440078310081, 22.3206294182717058396209474292, 22.38764911109133205020578441825

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