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Properties

Label 1-277-277.12-r0-0-0
Degree $1$
Conductor $277$
Sign $0.949 - 0.313i$
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  + (−0.203 − 0.979i)2-s + (−0.419 + 0.907i)3-s + (−0.917 + 0.398i)4-s + (−0.0227 − 0.999i)5-s + (0.974 + 0.225i)6-s + (−0.877 + 0.480i)7-s + (0.576 + 0.816i)8-s + (−0.648 − 0.761i)9-s + (−0.974 + 0.225i)10-s + (0.158 + 0.987i)11-s + (0.0227 − 0.999i)12-s + (−0.334 − 0.942i)13-s + (0.648 + 0.761i)14-s + (0.917 + 0.398i)15-s + (0.682 − 0.730i)16-s + (0.715 − 0.699i)17-s + ⋯
L(s)  = 1  + (−0.203 − 0.979i)2-s + (−0.419 + 0.907i)3-s + (−0.917 + 0.398i)4-s + (−0.0227 − 0.999i)5-s + (0.974 + 0.225i)6-s + (−0.877 + 0.480i)7-s + (0.576 + 0.816i)8-s + (−0.648 − 0.761i)9-s + (−0.974 + 0.225i)10-s + (0.158 + 0.987i)11-s + (0.0227 − 0.999i)12-s + (−0.334 − 0.942i)13-s + (0.648 + 0.761i)14-s + (0.917 + 0.398i)15-s + (0.682 − 0.730i)16-s + (0.715 − 0.699i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7288767888 - 0.1173858585i\)
\(L(\frac12)\) \(\approx\) \(0.7288767888 - 0.1173858585i\)
\(L(1)\) \(\approx\) \(0.7073304221 - 0.1530522156i\)
\(L(1)\) \(\approx\) \(0.7073304221 - 0.1530522156i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.203 - 0.979i)T \)
3 \( 1 + (-0.419 + 0.907i)T \)
5 \( 1 + (-0.0227 - 0.999i)T \)
7 \( 1 + (-0.877 + 0.480i)T \)
11 \( 1 + (0.158 + 0.987i)T \)
13 \( 1 + (-0.334 - 0.942i)T \)
17 \( 1 + (0.715 - 0.699i)T \)
19 \( 1 + (0.854 + 0.519i)T \)
23 \( 1 + (0.0227 + 0.999i)T \)
29 \( 1 + (0.113 + 0.993i)T \)
31 \( 1 + (0.877 + 0.480i)T \)
37 \( 1 + (0.917 - 0.398i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (0.949 + 0.313i)T \)
47 \( 1 + (-0.247 - 0.968i)T \)
53 \( 1 + (0.998 + 0.0455i)T \)
59 \( 1 + (-0.775 - 0.631i)T \)
61 \( 1 + (0.775 + 0.631i)T \)
67 \( 1 + (0.291 + 0.956i)T \)
71 \( 1 + (0.995 - 0.0909i)T \)
73 \( 1 + (-0.682 - 0.730i)T \)
79 \( 1 + (-0.877 - 0.480i)T \)
83 \( 1 + (-0.974 - 0.225i)T \)
89 \( 1 + (0.613 + 0.789i)T \)
97 \( 1 + (0.419 - 0.907i)T \)
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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

−25.89291308511008530053901313821, −24.62107128401409273335431952089, −24.035160391784259603425458509968, −23.03552391664824733312733754264, −22.53069468584344308951264016528, −21.6312344888885380451163738560, −19.58142602402574317839697782883, −19.031577790045210975230298742696, −18.4733005494279258282986288667, −17.280891432296927348962170621946, −16.663460436689560340053283554374, −15.71787208790381163956321262936, −14.301230660210422250800339790455, −13.88360624336199179974356816218, −12.88028143585306057589273941316, −11.58795427657164737023673366384, −10.50625502929758160901034096008, −9.467919503318784006323610845256, −8.08881285647452282190009802631, −7.231027066755674588174561869907, −6.396818720695667410445760982575, −5.87789959231630658505076662214, −4.156975685885951075436035682743, −2.74883477001962843711731556035, −0.79960422527327841992756766905, 0.97425205570950463172807167287, 2.79093083603364081179620583817, 3.8030648864506961011998014408, 4.98283740066340693180540415859, 5.628239788928544050192570912069, 7.62665504867809331161149656535, 8.96006894372440162148652173348, 9.672678006111267980372030630793, 10.175405057715755904204503210028, 11.68684293020237351290646924079, 12.27307226709984571427967688084, 13.04358780829571005025927573834, 14.436947342208436449366867562607, 15.70284493905545484788908475165, 16.4343885959194345680980612175, 17.433104399319673658632978091932, 18.17497511679637042768127828830, 19.66326426392837563488320958758, 20.1687274922326754067283581136, 21.01188042756505128540797116538, 21.8404402149923914813898867085, 22.81012722751714754736679099565, 23.20446058645333323967652133233, 24.968422885196673047240604645781, 25.70688380648347238710514615903

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