↑

Properties

Label 1-277-277.108-r0-0-0
Degree $1$
Conductor $277$
Sign $0.747 + 0.664i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.990 − 0.136i)2-s + (−0.917 + 0.398i)3-s + (0.962 − 0.269i)4-s + (0.775 + 0.631i)5-s + (−0.854 + 0.519i)6-s + (−0.775 + 0.631i)7-s + (0.917 − 0.398i)8-s + (0.682 − 0.730i)9-s + (0.854 + 0.519i)10-s + (0.0682 − 0.997i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (−0.682 + 0.730i)14-s + (−0.962 − 0.269i)15-s + (0.854 − 0.519i)16-s + (0.334 + 0.942i)17-s + ⋯
L(s)  = 1  + (0.990 − 0.136i)2-s + (−0.917 + 0.398i)3-s + (0.962 − 0.269i)4-s + (0.775 + 0.631i)5-s + (−0.854 + 0.519i)6-s + (−0.775 + 0.631i)7-s + (0.917 − 0.398i)8-s + (0.682 − 0.730i)9-s + (0.854 + 0.519i)10-s + (0.0682 − 0.997i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (−0.682 + 0.730i)14-s + (−0.962 − 0.269i)15-s + (0.854 − 0.519i)16-s + (0.334 + 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.767549057 + 0.6722023312i\)
\(L(\frac12)\) \(\approx\) \(1.767549057 + 0.6722023312i\)
\(L(1)\) \(\approx\) \(1.553395436 + 0.3020550198i\)
\(L(1)\) \(\approx\) \(1.553395436 + 0.3020550198i\)

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 \( 1 + (-0.917 + 0.398i)T \)
5 \( 1 + (0.775 + 0.631i)T \)
7 \( 1 + (-0.775 + 0.631i)T \)
11 \( 1 + (0.0682 - 0.997i)T \)
13 \( 1 + (0.682 + 0.730i)T \)
17 \( 1 + (0.334 + 0.942i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (-0.775 - 0.631i)T \)
29 \( 1 + (0.962 + 0.269i)T \)
31 \( 1 + (0.775 + 0.631i)T \)
37 \( 1 + (-0.962 + 0.269i)T \)
41 \( 1 + (-0.334 - 0.942i)T \)
43 \( 1 + (0.990 - 0.136i)T \)
47 \( 1 + (-0.334 - 0.942i)T \)
53 \( 1 + (-0.203 + 0.979i)T \)
59 \( 1 + (-0.0682 - 0.997i)T \)
61 \( 1 + (0.0682 + 0.997i)T \)
67 \( 1 + (0.854 - 0.519i)T \)
71 \( 1 + (-0.917 + 0.398i)T \)
73 \( 1 + (-0.854 - 0.519i)T \)
79 \( 1 + (-0.775 - 0.631i)T \)
83 \( 1 + (0.854 - 0.519i)T \)
89 \( 1 + (-0.576 - 0.816i)T \)
97 \( 1 + (0.917 - 0.398i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.35395383499532509508374159193, −24.52419295684207424220885354669, −23.499708283756426376448886178286, −22.923040167863313823801771784582, −22.21712977795368814159583194730, −21.14119662926874565704566503999, −20.34547700081146294751287800443, −19.37096809967780380061277748845, −17.77510168310246789530274273382, −17.27420434636962479887235864746, −16.21097104035892029620497003561, −15.62545762788672528107121179266, −14.01344945193205867087338161255, −13.23760241930041795608933335351, −12.68658508477145286455004008293, −11.76641107948914408807889970042, −10.5294573743778802268035306036, −9.75595955700736454178465324233, −7.90427592259340832695470299369, −6.792367101900497919788772308341, −6.09617222169774300435431915513, −5.08329755938469168964047535054, −4.1953433082896767176308176796, −2.54887416296090489225048731301, −1.17803666279840065630402988316, 1.69011351410693082796434460551, 3.12103158998006774868603010094, 4.07094955955944524913140142513, 5.55052949699586629525568835873, 6.20135518155602108599665492085, 6.66254300583472161426120877326, 8.72545345913912599418094029335, 10.22219364999881934495143317945, 10.610085846494523724923441294510, 11.8121566612118931180520355059, 12.56566007657232159283728942159, 13.66046311036339970946007750530, 14.52765513175383038502782910611, 15.65798714452032100172978490026, 16.32958523212284252119936955175, 17.2587386651487699554485376299, 18.66246930876652152369333833585, 19.18417999262128101944354079537, 20.90365793211054636182788198265, 21.5668463864704259596309610416, 22.010028630084833803000191125742, 22.90660183396612418834959142394, 23.66397975834941459150212229696, 24.6816857339991414902811426880, 25.71399500890148915099908354949

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