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Properties

Label 1-643-643.14-r1-0-0
Degree $1$
Conductor $643$
Sign $-0.551 - 0.834i$
Analytic cond. $69.0999$
Root an. cond. $69.0999$
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  + (−0.761 − 0.647i)2-s + (0.701 + 0.712i)3-s + (0.160 + 0.986i)4-s + (−0.541 + 0.840i)5-s + (−0.0733 − 0.997i)6-s + (−0.997 − 0.0684i)7-s + (0.516 − 0.856i)8-s + (−0.0146 + 0.999i)9-s + (0.957 − 0.289i)10-s + (−0.980 − 0.194i)11-s + (−0.590 + 0.807i)12-s + (0.208 + 0.977i)13-s + (0.715 + 0.698i)14-s + (−0.978 + 0.204i)15-s + (−0.948 + 0.317i)16-s + (0.170 + 0.985i)17-s + ⋯
L(s)  = 1  + (−0.761 − 0.647i)2-s + (0.701 + 0.712i)3-s + (0.160 + 0.986i)4-s + (−0.541 + 0.840i)5-s + (−0.0733 − 0.997i)6-s + (−0.997 − 0.0684i)7-s + (0.516 − 0.856i)8-s + (−0.0146 + 0.999i)9-s + (0.957 − 0.289i)10-s + (−0.980 − 0.194i)11-s + (−0.590 + 0.807i)12-s + (0.208 + 0.977i)13-s + (0.715 + 0.698i)14-s + (−0.978 + 0.204i)15-s + (−0.948 + 0.317i)16-s + (0.170 + 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ -0.551 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2405046093 + 0.4470439170i\)
\(L(\frac12)\) \(\approx\) \(-0.2405046093 + 0.4470439170i\)
\(L(1)\) \(\approx\) \(0.5633681009 + 0.2787153540i\)
\(L(1)\) \(\approx\) \(0.5633681009 + 0.2787153540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.761 - 0.647i)T \)
3 \( 1 + (0.701 + 0.712i)T \)
5 \( 1 + (-0.541 + 0.840i)T \)
7 \( 1 + (-0.997 - 0.0684i)T \)
11 \( 1 + (-0.980 - 0.194i)T \)
13 \( 1 + (0.208 + 0.977i)T \)
17 \( 1 + (0.170 + 0.985i)T \)
19 \( 1 + (-0.256 + 0.966i)T \)
23 \( 1 + (-0.0928 + 0.995i)T \)
29 \( 1 + (-0.322 - 0.946i)T \)
31 \( 1 + (0.312 + 0.949i)T \)
37 \( 1 + (-0.810 + 0.586i)T \)
41 \( 1 + (0.112 - 0.993i)T \)
43 \( 1 + (0.613 + 0.789i)T \)
47 \( 1 + (-0.558 + 0.829i)T \)
53 \( 1 + (-0.533 - 0.845i)T \)
59 \( 1 + (0.265 + 0.964i)T \)
61 \( 1 + (0.284 + 0.958i)T \)
67 \( 1 + (-0.891 + 0.452i)T \)
71 \( 1 + (-0.984 + 0.175i)T \)
73 \( 1 + (0.687 - 0.725i)T \)
79 \( 1 + (0.340 + 0.940i)T \)
83 \( 1 + (0.312 - 0.949i)T \)
89 \( 1 + (-0.941 - 0.335i)T \)
97 \( 1 + (0.976 - 0.213i)T \)
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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

−22.53687190768982950924834272553, −20.769726400413867479416043808192, −20.258702672921511300882980331096, −19.63334421089715560266457294886, −18.77019794542605181390831020179, −18.19698311920475815034965646813, −17.23590484059983866582241061257, −16.19069937897164387350792937730, −15.64307066245047384918750953145, −14.950357664054906985803584121005, −13.68213165680392244548167032612, −13.00102553272690101657855401310, −12.29600550299477705130469348707, −10.982329221454282309256191687740, −9.86180540700586214841407532832, −9.08746986544214491479118369402, −8.3592637213595248949817492218, −7.582394140789021977883641015683, −6.87098049797789371626736233550, −5.76591157701193387912935543988, −4.77598364042158314611374997955, −3.232040537366343419491677374891, −2.25752488776738195953507491715, −0.7128939563483580614443980698, −0.20326286526354068394024296762, 1.80886454049927724333033162486, 2.874225744254900158059108190171, 3.54505288419437359314253421607, 4.24244717516717268025158498776, 6.02448667653008756051893011619, 7.20194610934946067734560488771, 7.98677497473989813056277300626, 8.79424956292267440184234554486, 9.86689701289048819958712387813, 10.32392485644275895246814940337, 11.10016645462131013537004034325, 12.12766894000178931074209740344, 13.17583043294891245689830411022, 13.98049691318346646163362003721, 15.131173000288576341203971511631, 15.9470324773907836120846314099, 16.39259011090750540154581408763, 17.55722544403454862650446965024, 18.780317813126082171831455873119, 19.20991132920381140775116923180, 19.60739336555848160402134268423, 20.921472469599428602297113774292, 21.25204329723498218072669656739, 22.23457566029988195234724470745, 22.90592385953139890406982262400

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