↑

Properties

Label 1-277-277.191-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.364 + 0.931i$
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  + (0.854 + 0.519i)2-s + (−0.829 − 0.557i)3-s + (0.460 + 0.887i)4-s + (0.113 + 0.993i)5-s + (−0.419 − 0.907i)6-s + (0.803 + 0.595i)7-s + (−0.0682 + 0.997i)8-s + (0.377 + 0.926i)9-s + (−0.419 + 0.907i)10-s + (−0.715 − 0.699i)11-s + (0.113 − 0.993i)12-s + (−0.990 + 0.136i)13-s + (0.377 + 0.926i)14-s + (0.460 − 0.887i)15-s + (−0.576 + 0.816i)16-s + (0.746 − 0.665i)17-s + ⋯
L(s)  = 1  + (0.854 + 0.519i)2-s + (−0.829 − 0.557i)3-s + (0.460 + 0.887i)4-s + (0.113 + 0.993i)5-s + (−0.419 − 0.907i)6-s + (0.803 + 0.595i)7-s + (−0.0682 + 0.997i)8-s + (0.377 + 0.926i)9-s + (−0.419 + 0.907i)10-s + (−0.715 − 0.699i)11-s + (0.113 − 0.993i)12-s + (−0.990 + 0.136i)13-s + (0.377 + 0.926i)14-s + (0.460 − 0.887i)15-s + (−0.576 + 0.816i)16-s + (0.746 − 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.364 + 0.931i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.364 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8324927795 + 1.220176326i\)
\(L(\frac12)\) \(\approx\) \(0.8324927795 + 1.220176326i\)
\(L(1)\) \(\approx\) \(1.133992793 + 0.6623809372i\)
\(L(1)\) \(\approx\) \(1.133992793 + 0.6623809372i\)

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 \( 1 + (-0.829 - 0.557i)T \)
5 \( 1 + (0.113 + 0.993i)T \)
7 \( 1 + (0.803 + 0.595i)T \)
11 \( 1 + (-0.715 - 0.699i)T \)
13 \( 1 + (-0.990 + 0.136i)T \)
17 \( 1 + (0.746 - 0.665i)T \)
19 \( 1 + (-0.917 + 0.398i)T \)
23 \( 1 + (0.113 + 0.993i)T \)
29 \( 1 + (0.538 + 0.842i)T \)
31 \( 1 + (0.803 - 0.595i)T \)
37 \( 1 + (0.460 + 0.887i)T \)
41 \( 1 + (0.203 + 0.979i)T \)
43 \( 1 + (0.0227 - 0.999i)T \)
47 \( 1 + (-0.949 - 0.313i)T \)
53 \( 1 + (-0.974 - 0.225i)T \)
59 \( 1 + (0.962 + 0.269i)T \)
61 \( 1 + (0.962 + 0.269i)T \)
67 \( 1 + (0.995 + 0.0909i)T \)
71 \( 1 + (0.898 - 0.439i)T \)
73 \( 1 + (-0.576 - 0.816i)T \)
79 \( 1 + (0.803 - 0.595i)T \)
83 \( 1 + (-0.419 - 0.907i)T \)
89 \( 1 + (-0.158 - 0.987i)T \)
97 \( 1 + (-0.829 - 0.557i)T \)
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

−24.987039634779725814183940052693, −24.08725625060659737610853425400, −23.47722923415321815500714209918, −22.76889705064534038056341493700, −21.462659267151569102322158984703, −21.09029128521910405618114465239, −20.32409541653601174947717905675, −19.29000027740756875493542583222, −17.74748018038529795900443831958, −17.10108796180297498978168495710, −16.070062590506908698046647173028, −15.09715920114383954441155463208, −14.263736777050605781787092663281, −12.77140255343157052121201397695, −12.45853560761521591432995257986, −11.31177136598600205387531061288, −10.35705443264545281534310296890, −9.72027287251796874268702254071, −8.0865680737808438227696740182, −6.675948961531461195109925071513, −5.3943759572102594755800713736, −4.73107835902008117225173840928, −4.11504190335774058212124593822, −2.25689461029169815130228563917, −0.840364186392037852954013348482, 2.05360203005385314899927782816, 3.04729070171444547046187400862, 4.77885418599083070736804366298, 5.54800337590606891826516156527, 6.4490167613852407882949274094, 7.47441632628925325270015901862, 8.17911523721524200968504196010, 10.16565224090520020129326597270, 11.3292070149463565627374136956, 11.79456108756458440995278367644, 12.906108060638631324477571542998, 13.93088787726580175083098910833, 14.71188969384800252618026586996, 15.658122848972763926152329418261, 16.74745750496111092901665202603, 17.61726065026336961736450628685, 18.43046893142381966071992598689, 19.28831061831899894533225092898, 21.03609816597221906625241023934, 21.695202756826929691885156048291, 22.34424387939106289878949768585, 23.40499503743656857008815929707, 23.8697091481317230934465140784, 24.91026039396767945238521975129, 25.55129933279164485941969755900

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