Properties

Label 1-47e2-2209.1020-r1-0-0
Degree $1$
Conductor $2209$
Sign $0.690 - 0.722i$
Analytic cond. $237.390$
Root an. cond. $237.390$
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  + (0.981 − 0.190i)2-s + (−0.843 − 0.536i)3-s + (0.927 − 0.374i)4-s + (0.940 + 0.338i)5-s + (−0.930 − 0.366i)6-s + (0.802 + 0.596i)7-s + (0.838 − 0.544i)8-s + (0.423 + 0.905i)9-s + (0.988 + 0.153i)10-s + (−0.525 − 0.850i)11-s + (−0.983 − 0.182i)12-s + (0.998 + 0.0609i)13-s + (0.901 + 0.432i)14-s + (−0.611 − 0.791i)15-s + (0.719 − 0.694i)16-s + (0.976 − 0.213i)17-s + ⋯
L(s)  = 1  + (0.981 − 0.190i)2-s + (−0.843 − 0.536i)3-s + (0.927 − 0.374i)4-s + (0.940 + 0.338i)5-s + (−0.930 − 0.366i)6-s + (0.802 + 0.596i)7-s + (0.838 − 0.544i)8-s + (0.423 + 0.905i)9-s + (0.988 + 0.153i)10-s + (−0.525 − 0.850i)11-s + (−0.983 − 0.182i)12-s + (0.998 + 0.0609i)13-s + (0.901 + 0.432i)14-s + (−0.611 − 0.791i)15-s + (0.719 − 0.694i)16-s + (0.976 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.690 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.690 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $0.690 - 0.722i$
Analytic conductor: \(237.390\)
Root analytic conductor: \(237.390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (1020, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2209,\ (1:\ ),\ 0.690 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.799273442 - 2.479331279i\)
\(L(\frac12)\) \(\approx\) \(5.799273442 - 2.479331279i\)
\(L(1)\) \(\approx\) \(2.290586197 - 0.5928709083i\)
\(L(1)\) \(\approx\) \(2.290586197 - 0.5928709083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (0.981 - 0.190i)T \)
3 \( 1 + (-0.843 - 0.536i)T \)
5 \( 1 + (0.940 + 0.338i)T \)
7 \( 1 + (0.802 + 0.596i)T \)
11 \( 1 + (-0.525 - 0.850i)T \)
13 \( 1 + (0.998 + 0.0609i)T \)
17 \( 1 + (0.976 - 0.213i)T \)
19 \( 1 + (0.733 - 0.679i)T \)
23 \( 1 + (0.924 + 0.382i)T \)
29 \( 1 + (-0.0943 - 0.995i)T \)
31 \( 1 + (0.986 - 0.164i)T \)
37 \( 1 + (0.883 + 0.469i)T \)
41 \( 1 + (-0.397 + 0.917i)T \)
43 \( 1 + (0.436 - 0.899i)T \)
53 \( 1 + (-0.334 + 0.942i)T \)
59 \( 1 + (-0.814 + 0.580i)T \)
61 \( 1 + (-0.772 + 0.635i)T \)
67 \( 1 + (-0.203 - 0.979i)T \)
71 \( 1 + (-0.917 + 0.398i)T \)
73 \( 1 + (0.102 - 0.994i)T \)
79 \( 1 + (-0.952 - 0.305i)T \)
83 \( 1 + (0.885 + 0.463i)T \)
89 \( 1 + (0.111 - 0.993i)T \)
97 \( 1 + (-0.907 - 0.419i)T \)
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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

−20.28104215488150948651118060859, −18.70851952636608156684371780473, −17.84277631982096032252333532669, −17.41266254636608726368947755410, −16.55475152565938489760159217607, −16.20109443750828281776030336451, −15.22884778679397526452973569692, −14.50789444426387733116316351917, −13.937482237072279213842310402585, −12.98322086977945500884333050840, −12.51101073954602398770012705666, −11.662730604758828369540088478085, −10.79793138160648535761312729049, −10.38855149254627668569655264199, −9.564745903174775472173719145964, −8.37529080734890587103667414512, −7.4842083733607606487723235066, −6.649966439154376339398704070781, −5.839141937414333292579831047980, −5.21293618344482894070790887409, −4.70784466103986621851127596300, −3.8619386501185985820808064197, −2.91201653533254785331661260788, −1.56648321318542721963310406751, −1.09350290940901615536658659984, 0.97041219368702561410412613389, 1.435937429682775213485396649816, 2.57303853920671550025037683443, 3.10662989298874523442416393079, 4.57262454162713625063735993799, 5.23513975194022309372695941256, 5.923454591815873940442601986784, 6.20571701504228895364442353150, 7.33932077205706231697032084715, 8.008887720475289531238052738388, 9.22215090439962885085560006145, 10.28092246276596569716203145553, 10.920932409050120906915757092030, 11.503720914321446332993188449754, 12.039447316581042694676689387629, 13.13393789621134811605699563945, 13.56497693327771805127139179736, 14.02727526102773625538697781960, 15.10275301887501675907106386634, 15.71632631286425349600041555608, 16.62191100775187790127402895203, 17.23136687227481295616117367929, 18.223428141388675253075188697453, 18.60438485082637938051921210119, 19.256150782918141399342490311978

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