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Properties

Label 1-277-277.188-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.995 - 0.0937i$
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  + (−0.334 − 0.942i)2-s + (0.0227 − 0.999i)3-s + (−0.775 + 0.631i)4-s + (0.613 − 0.789i)5-s + (−0.949 + 0.313i)6-s + (0.377 − 0.926i)7-s + (0.854 + 0.519i)8-s + (−0.998 − 0.0455i)9-s + (−0.949 − 0.313i)10-s + (0.995 − 0.0909i)11-s + (0.613 + 0.789i)12-s + (0.460 − 0.887i)13-s + (−0.998 − 0.0455i)14-s + (−0.775 − 0.631i)15-s + (0.203 − 0.979i)16-s + (0.898 + 0.439i)17-s + ⋯
L(s)  = 1  + (−0.334 − 0.942i)2-s + (0.0227 − 0.999i)3-s + (−0.775 + 0.631i)4-s + (0.613 − 0.789i)5-s + (−0.949 + 0.313i)6-s + (0.377 − 0.926i)7-s + (0.854 + 0.519i)8-s + (−0.998 − 0.0455i)9-s + (−0.949 − 0.313i)10-s + (0.995 − 0.0909i)11-s + (0.613 + 0.789i)12-s + (0.460 − 0.887i)13-s + (−0.998 − 0.0455i)14-s + (−0.775 − 0.631i)15-s + (0.203 − 0.979i)16-s + (0.898 + 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.995 - 0.0937i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.995 - 0.0937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05402654424 - 1.150372230i\)
\(L(\frac12)\) \(\approx\) \(0.05402654424 - 1.150372230i\)
\(L(1)\) \(\approx\) \(0.5582599659 - 0.8412751555i\)
\(L(1)\) \(\approx\) \(0.5582599659 - 0.8412751555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.334 - 0.942i)T \)
3 \( 1 + (0.0227 - 0.999i)T \)
5 \( 1 + (0.613 - 0.789i)T \)
7 \( 1 + (0.377 - 0.926i)T \)
11 \( 1 + (0.995 - 0.0909i)T \)
13 \( 1 + (0.460 - 0.887i)T \)
17 \( 1 + (0.898 + 0.439i)T \)
19 \( 1 + (-0.990 + 0.136i)T \)
23 \( 1 + (0.613 - 0.789i)T \)
29 \( 1 + (-0.158 + 0.987i)T \)
31 \( 1 + (0.377 + 0.926i)T \)
37 \( 1 + (-0.775 + 0.631i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (0.983 + 0.181i)T \)
47 \( 1 + (-0.829 + 0.557i)T \)
53 \( 1 + (-0.247 + 0.968i)T \)
59 \( 1 + (-0.576 + 0.816i)T \)
61 \( 1 + (-0.576 + 0.816i)T \)
67 \( 1 + (0.746 + 0.665i)T \)
71 \( 1 + (-0.877 + 0.480i)T \)
73 \( 1 + (0.203 + 0.979i)T \)
79 \( 1 + (0.377 + 0.926i)T \)
83 \( 1 + (-0.949 + 0.313i)T \)
89 \( 1 + (0.291 - 0.956i)T \)
97 \( 1 + (0.0227 - 0.999i)T \)
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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

−25.97546333731652251741627668814, −25.41036930822654301676449099467, −24.62725559875266786143070131778, −23.21833718054259887854883812456, −22.55187569636124686134679243158, −21.6096588344039433488978412019, −21.05759450390047730503640673483, −19.332159571346318626031799199101, −18.730704683479921248922467993744, −17.53411188751635427133157476176, −16.95783604863491115970847057896, −15.87291761342778481827490473228, −14.91865919093588623631314608987, −14.50310628779095589665465225483, −13.559274355521356022753296711338, −11.72862445733358143319873569419, −10.828690736937941254368744078265, −9.51257307328237773137880010494, −9.25660448538435742991463465227, −8.02013992383593295224220071171, −6.548409081385621096884872070012, −5.864695086271930063931311336325, −4.78932539045793471863207285211, −3.57389992514305448003123902046, −1.93767664063804089827193882501, 1.02391271393612656758585841850, 1.54748337865426783887605882431, 3.09157137073364737820601709020, 4.377770895427380624321529612121, 5.68846765974611623465595524021, 7.03336785117062699048271536741, 8.293983232670346739475124236555, 8.80370748759894409288686740748, 10.21851950565812028713339723568, 11.02515850306186346652807003328, 12.354910034236119079530711691419, 12.75567523393858412729277054796, 13.80512738112517900336426238819, 14.43248053066582000736813450618, 16.660672303007454968787316925970, 17.19111134457177779019285240422, 17.836914146480016006876255629922, 18.96194642520947113147789003706, 19.79610339121326945640722950349, 20.51664738120153669536032937928, 21.24159949904758133397170399141, 22.558843477389408683213218742579, 23.335644140479291541315371144185, 24.32038937391818995805495570213, 25.33635611969291443839461378796

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