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Properties

Label 1-277-277.171-r0-0-0
Degree $1$
Conductor $277$
Sign $0.870 + 0.492i$
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  + (−0.990 + 0.136i)2-s + (0.113 − 0.993i)3-s + (0.962 − 0.269i)4-s + (−0.158 + 0.987i)5-s + (0.0227 + 0.999i)6-s + (0.934 + 0.356i)7-s + (−0.917 + 0.398i)8-s + (−0.974 − 0.225i)9-s + (0.0227 − 0.999i)10-s + (0.898 − 0.439i)11-s + (−0.158 − 0.987i)12-s + (0.682 + 0.730i)13-s + (−0.974 − 0.225i)14-s + (0.962 + 0.269i)15-s + (0.854 − 0.519i)16-s + (−0.648 + 0.761i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.136i)2-s + (0.113 − 0.993i)3-s + (0.962 − 0.269i)4-s + (−0.158 + 0.987i)5-s + (0.0227 + 0.999i)6-s + (0.934 + 0.356i)7-s + (−0.917 + 0.398i)8-s + (−0.974 − 0.225i)9-s + (0.0227 − 0.999i)10-s + (0.898 − 0.439i)11-s + (−0.158 − 0.987i)12-s + (0.682 + 0.730i)13-s + (−0.974 − 0.225i)14-s + (0.962 + 0.269i)15-s + (0.854 − 0.519i)16-s + (−0.648 + 0.761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.870 + 0.492i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.870 + 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8321234673 + 0.2189546800i\)
\(L(\frac12)\) \(\approx\) \(0.8321234673 + 0.2189546800i\)
\(L(1)\) \(\approx\) \(0.7865894965 + 0.04131193059i\)
\(L(1)\) \(\approx\) \(0.7865894965 + 0.04131193059i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.990 + 0.136i)T \)
3 \( 1 + (0.113 - 0.993i)T \)
5 \( 1 + (-0.158 + 0.987i)T \)
7 \( 1 + (0.934 + 0.356i)T \)
11 \( 1 + (0.898 - 0.439i)T \)
13 \( 1 + (0.682 + 0.730i)T \)
17 \( 1 + (-0.648 + 0.761i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (-0.158 + 0.987i)T \)
29 \( 1 + (-0.715 + 0.699i)T \)
31 \( 1 + (0.934 - 0.356i)T \)
37 \( 1 + (0.962 - 0.269i)T \)
41 \( 1 + (-0.334 - 0.942i)T \)
43 \( 1 + (0.613 + 0.789i)T \)
47 \( 1 + (0.983 + 0.181i)T \)
53 \( 1 + (-0.949 + 0.313i)T \)
59 \( 1 + (-0.0682 - 0.997i)T \)
61 \( 1 + (-0.0682 - 0.997i)T \)
67 \( 1 + (-0.877 - 0.480i)T \)
71 \( 1 + (0.803 + 0.595i)T \)
73 \( 1 + (0.854 + 0.519i)T \)
79 \( 1 + (0.934 - 0.356i)T \)
83 \( 1 + (0.0227 + 0.999i)T \)
89 \( 1 + (0.995 - 0.0909i)T \)
97 \( 1 + (0.113 - 0.993i)T \)
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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

−25.62792168649733502911428208959, −24.936183545048841846038828348491, −24.0543937866519234048236048479, −22.788729383301894926625597234730, −21.5912324413128648975248964681, −20.55026089611374020367375842709, −20.420632361506611229685463419, −19.47584260324925198160077928561, −17.96481968755360620149252380417, −17.2196080896071451970372048086, −16.57658420326100631781941527839, −15.561437234451913096747183378027, −14.87086787258766841716061438753, −13.46532901484036771456992234706, −12.02977482003815160266114248887, −11.244656682663399293244260892553, −10.377736646038202297596236165421, −9.23488008339165962970137407553, −8.63528196588564387183251283385, −7.77170670134332639931903383613, −6.24134492634782727488933560287, −4.80555394538094686418733522726, −3.98399495925409235532962068975, −2.35896276139992705039500908110, −0.866254506379996138244752504732, 1.46628874738736198463953367328, 2.23363790411664403933838624688, 3.70838298973255879574438129418, 5.97887911548827105404222240926, 6.49504624937737495902180445431, 7.61276341938040240304344754464, 8.38492962218119557275119157297, 9.2855544452731125832635236033, 10.996172956575763703550351800965, 11.2710595724588510383454078047, 12.30495247416793786916278675814, 13.93174652680363978253289473592, 14.59817201526060375824373831940, 15.49813256398910323873383533068, 16.97152878897690595194540529264, 17.6138528140047181260835067313, 18.54808009995213551085460305390, 19.02827743063185369142280157165, 19.83492909236254428378160210829, 21.03766475670468862172264428027, 22.037430717944167496772948272209, 23.47433298016191929245174465333, 24.02737074230419333085683204880, 24.993997917527020765370617188768, 25.71920482420684544327613098385

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