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Properties

Label 1-277-277.160-r0-0-0
Degree $1$
Conductor $277$
Sign $0.680 - 0.732i$
Analytic cond. $1.28638$
Root an. cond. $1.28638$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + (−0.5 − 0.866i)14-s + 15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + (−0.5 − 0.866i)14-s + 15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.680 - 0.732i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.680 - 0.732i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.501221350 - 0.6546243301i\)
\(L(\frac12)\) \(\approx\) \(1.501221350 - 0.6546243301i\)
\(L(1)\) \(\approx\) \(1.425157942 - 0.1867061299i\)
\(L(1)\) \(\approx\) \(1.425157942 - 0.1867061299i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−25.61869554437778748178069547922, −24.69279825724113646843061904497, −23.69674208140917026556578556008, −23.063660219365231966806510863934, −22.38331141284703699313393941809, −21.65217058859529894163413739251, −20.263776723104144572224799309150, −19.38305976406267697431767055864, −18.51413694198908224856583551980, −17.71921049176577369747599297983, −16.21401449238288156512364216249, −15.49892698243610023839426260284, −14.65047039940400128436896159211, −13.37689133778500854264470132330, −12.82146051491668631922285424455, −11.679638654526174161753968056640, −11.25481195921911118997451351940, −9.97337416369804531972038978653, −8.05942824530421487272796524196, −7.253424009775426062905552279507, −6.21078515092626881389542413157, −5.63116368645558295128768549944, −4.070516384821233528816572989889, −2.83787379198546941377499577494, −1.875667721594299739985306700240, 0.87653901500864002416180151238, 3.18088954373292006616127503087, 3.91808919655850436266462491970, 4.89458363733678136826927598642, 5.766038205233830645309556989444, 6.922853452899111762075926127, 8.26405551274000369827185954639, 9.53984586832269572121548289410, 10.82886603655894688921806588460, 11.32341116748564766662413533074, 12.49226405903409410710726421170, 13.397702253631855227022172942368, 14.28122128207181327798573505558, 15.69109729988968964775781946846, 16.27551267519338685776389103328, 16.51236649309262499129503354506, 18.07613700116092152507617529176, 19.65916431270862942703730408042, 20.48829746919903318817665558625, 20.87598180161160883670944857576, 22.09332668752880700230860055458, 22.79844282444151413459809780033, 23.65076821007948494790477116964, 24.17281648513357557220084596697, 25.47729871108252794444369526506

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