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Properties

Label 1-277-277.131-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.263 + 0.964i$
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  + (0.682 − 0.730i)2-s + (−0.775 − 0.631i)3-s + (−0.0682 − 0.997i)4-s + (−0.576 − 0.816i)5-s + (−0.990 + 0.136i)6-s + (−0.576 + 0.816i)7-s + (−0.775 − 0.631i)8-s + (0.203 + 0.979i)9-s + (−0.990 − 0.136i)10-s + (−0.917 − 0.398i)11-s + (−0.576 + 0.816i)12-s + (0.203 − 0.979i)13-s + (0.203 + 0.979i)14-s + (−0.0682 + 0.997i)15-s + (−0.990 + 0.136i)16-s + (0.460 + 0.887i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.730i)2-s + (−0.775 − 0.631i)3-s + (−0.0682 − 0.997i)4-s + (−0.576 − 0.816i)5-s + (−0.990 + 0.136i)6-s + (−0.576 + 0.816i)7-s + (−0.775 − 0.631i)8-s + (0.203 + 0.979i)9-s + (−0.990 − 0.136i)10-s + (−0.917 − 0.398i)11-s + (−0.576 + 0.816i)12-s + (0.203 − 0.979i)13-s + (0.203 + 0.979i)14-s + (−0.0682 + 0.997i)15-s + (−0.990 + 0.136i)16-s + (0.460 + 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.263 + 0.964i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.263 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2101301095 - 0.2751623869i\)
\(L(\frac12)\) \(\approx\) \(-0.2101301095 - 0.2751623869i\)
\(L(1)\) \(\approx\) \(0.4874220181 - 0.5315781891i\)
\(L(1)\) \(\approx\) \(0.4874220181 - 0.5315781891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (0.682 - 0.730i)T \)
3 \( 1 + (-0.775 - 0.631i)T \)
5 \( 1 + (-0.576 - 0.816i)T \)
7 \( 1 + (-0.576 + 0.816i)T \)
11 \( 1 + (-0.917 - 0.398i)T \)
13 \( 1 + (0.203 - 0.979i)T \)
17 \( 1 + (0.460 + 0.887i)T \)
19 \( 1 + (-0.576 + 0.816i)T \)
23 \( 1 + (-0.576 - 0.816i)T \)
29 \( 1 + (-0.0682 + 0.997i)T \)
31 \( 1 + (-0.576 - 0.816i)T \)
37 \( 1 + (-0.0682 - 0.997i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (0.682 - 0.730i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (-0.334 - 0.942i)T \)
59 \( 1 + (-0.917 + 0.398i)T \)
61 \( 1 + (-0.917 + 0.398i)T \)
67 \( 1 + (-0.990 + 0.136i)T \)
71 \( 1 + (-0.775 - 0.631i)T \)
73 \( 1 + (-0.990 - 0.136i)T \)
79 \( 1 + (-0.576 - 0.816i)T \)
83 \( 1 + (-0.990 + 0.136i)T \)
89 \( 1 + (0.854 - 0.519i)T \)
97 \( 1 + (-0.775 - 0.631i)T \)
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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

−26.221239166632476017981406985684, −25.720701291077311372369111764488, −23.988254616259010248359830262226, −23.35755476781604287360187049360, −22.963893153329985477964891250000, −21.99957327108938531038050149578, −21.21817670857379633938908853050, −20.18639817249329467443472069111, −18.77720511758737047579847762103, −17.77361013884811202173421051902, −16.8649431996408229791300804759, −15.883984447816873327555276150826, −15.53840392524170650346688283824, −14.33338825205699687852074941297, −13.43361794866952198959678761679, −12.207595755421464217251678546980, −11.35843342986451062554127030337, −10.41050304590309695410288540439, −9.26984096253252910733630615893, −7.596508032593113412217062637661, −6.9396499103162267455398977879, −6.011198491622354019757416241729, −4.70516542039034842161965329283, −3.94565079608782024559145568174, −2.888421714245254983423054597465, 0.19826052033849481303508752324, 1.69452966465175737818339505781, 3.03656581526420372578260941988, 4.37792180665811383568716135318, 5.70034887049174550893312211107, 5.89323771701660584643396995003, 7.70757040562377311851372449541, 8.742063922622684711078516887952, 10.25657176334868169747064736417, 11.00215210378523259760705154518, 12.268290117685267622785505184559, 12.61283767482362452833228793259, 13.225946439832089744875829855781, 14.75224706170926402765296630854, 15.83115467349404803753018123412, 16.50710747569162210846545465716, 17.993651549659323812095378256087, 18.80260727195607795097831605309, 19.467140603578888243937976091447, 20.536073951475571374268722994310, 21.471610050325688913019474051995, 22.41918842875987664243684107891, 23.17791239452315554485580147231, 23.912360937507963002963010458306, 24.61281388023305812531322127729

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