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Properties

Label 1-277-277.187-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.405 - 0.914i$
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  + (−0.854 − 0.519i)2-s + (0.898 − 0.439i)3-s + (0.460 + 0.887i)4-s + (−0.803 + 0.595i)5-s + (−0.995 − 0.0909i)6-s + (0.113 − 0.993i)7-s + (0.0682 − 0.997i)8-s + (0.613 − 0.789i)9-s + (0.995 − 0.0909i)10-s + (0.247 − 0.968i)11-s + (0.803 + 0.595i)12-s + (−0.990 + 0.136i)13-s + (−0.613 + 0.789i)14-s + (−0.460 + 0.887i)15-s + (−0.576 + 0.816i)16-s + (0.949 + 0.313i)17-s + ⋯
L(s)  = 1  + (−0.854 − 0.519i)2-s + (0.898 − 0.439i)3-s + (0.460 + 0.887i)4-s + (−0.803 + 0.595i)5-s + (−0.995 − 0.0909i)6-s + (0.113 − 0.993i)7-s + (0.0682 − 0.997i)8-s + (0.613 − 0.789i)9-s + (0.995 − 0.0909i)10-s + (0.247 − 0.968i)11-s + (0.803 + 0.595i)12-s + (−0.990 + 0.136i)13-s + (−0.613 + 0.789i)14-s + (−0.460 + 0.887i)15-s + (−0.576 + 0.816i)16-s + (0.949 + 0.313i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.405 - 0.914i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.405 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4769719957 - 0.7333208228i\)
\(L(\frac12)\) \(\approx\) \(0.4769719957 - 0.7333208228i\)
\(L(1)\) \(\approx\) \(0.7236857454 - 0.4075525646i\)
\(L(1)\) \(\approx\) \(0.7236857454 - 0.4075525646i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.854 - 0.519i)T \)
3 \( 1 + (0.898 - 0.439i)T \)
5 \( 1 + (-0.803 + 0.595i)T \)
7 \( 1 + (0.113 - 0.993i)T \)
11 \( 1 + (0.247 - 0.968i)T \)
13 \( 1 + (-0.990 + 0.136i)T \)
17 \( 1 + (0.949 + 0.313i)T \)
19 \( 1 + (-0.917 + 0.398i)T \)
23 \( 1 + (0.803 - 0.595i)T \)
29 \( 1 + (-0.998 + 0.0455i)T \)
31 \( 1 + (-0.113 - 0.993i)T \)
37 \( 1 + (-0.460 - 0.887i)T \)
41 \( 1 + (0.203 + 0.979i)T \)
43 \( 1 + (0.877 - 0.480i)T \)
47 \( 1 + (0.746 - 0.665i)T \)
53 \( 1 + (-0.291 - 0.956i)T \)
59 \( 1 + (0.962 + 0.269i)T \)
61 \( 1 + (-0.962 - 0.269i)T \)
67 \( 1 + (-0.419 - 0.907i)T \)
71 \( 1 + (-0.829 - 0.557i)T \)
73 \( 1 + (0.576 + 0.816i)T \)
79 \( 1 + (0.113 + 0.993i)T \)
83 \( 1 + (0.995 + 0.0909i)T \)
89 \( 1 + (0.934 + 0.356i)T \)
97 \( 1 + (-0.898 + 0.439i)T \)
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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

−25.78651435400671838105029031878, −25.20450895797806826392361516534, −24.504423351298330079423330475947, −23.54522541883913129921849336611, −22.34216037999683091004637219579, −21.02534417376817334813420675892, −20.34499279277825880886112729526, −19.271348194113747803461622985560, −19.03845406081000530209861513090, −17.58112189429845449446919635589, −16.66189674221463779738866886408, −15.605114074999943925051938682693, −15.109330757606224812548817680755, −14.41949951153926612944738129351, −12.74654845302204206924885559520, −11.840460403941058221823049555669, −10.515948468398913650129930572584, −9.33665054006685020715881034970, −8.94626879761422466349085909508, −7.80639351756019414267513502814, −7.18695845236254423511301077577, −5.37040726121039114157875187166, −4.55181424525612147592828874575, −2.88064020869592096185016478039, −1.672119613370973363164330545922, 0.71873488467849634576811761201, 2.23351143391296049985429382756, 3.404323483338230346217476712317, 4.05236322681596444829634127520, 6.5454921992418507672835084525, 7.48302518258662500460943873179, 7.99724330825941342794487624422, 9.09216147096345003804391903206, 10.242350522028511797536724063822, 11.05173366712141083393826060634, 12.149334206908977257999391969488, 13.06602125045570855301870388742, 14.313159635747529808581446361921, 15.01832659641222993662560996954, 16.436151186119871091907309974487, 17.116770807543683583547842562828, 18.46523593977698093188977482455, 19.17662534683910184100040092143, 19.56364771183933682219316676976, 20.580929750649266346641185285530, 21.364361679099885640755951782234, 22.603007566034840984761501186549, 23.79474477567504265080748337072, 24.55767198374398863994460097009, 25.70763602505360668667194910887

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