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Properties

Label 1-277-277.102-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.576 + 0.817i$
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 + (−0.576 + 0.816i)3-s + (−0.917 − 0.398i)4-s + (−0.854 − 0.519i)5-s + (−0.682 − 0.730i)6-s + (0.854 − 0.519i)7-s + (0.576 − 0.816i)8-s + (−0.334 − 0.942i)9-s + (0.682 − 0.730i)10-s + (0.775 + 0.631i)11-s + (0.854 − 0.519i)12-s + (−0.334 + 0.942i)13-s + (0.334 + 0.942i)14-s + (0.917 − 0.398i)15-s + (0.682 + 0.730i)16-s + (−0.962 + 0.269i)17-s + ⋯
L(s)  = 1  + (−0.203 + 0.979i)2-s + (−0.576 + 0.816i)3-s + (−0.917 − 0.398i)4-s + (−0.854 − 0.519i)5-s + (−0.682 − 0.730i)6-s + (0.854 − 0.519i)7-s + (0.576 − 0.816i)8-s + (−0.334 − 0.942i)9-s + (0.682 − 0.730i)10-s + (0.775 + 0.631i)11-s + (0.854 − 0.519i)12-s + (−0.334 + 0.942i)13-s + (0.334 + 0.942i)14-s + (0.917 − 0.398i)15-s + (0.682 + 0.730i)16-s + (−0.962 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.576 + 0.817i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.576 + 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3283255218 + 0.6334619365i\)
\(L(\frac12)\) \(\approx\) \(0.3283255218 + 0.6334619365i\)
\(L(1)\) \(\approx\) \(0.5582931730 + 0.4349603036i\)
\(L(1)\) \(\approx\) \(0.5582931730 + 0.4349603036i\)

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 \( 1 + (-0.576 + 0.816i)T \)
5 \( 1 + (-0.854 - 0.519i)T \)
7 \( 1 + (0.854 - 0.519i)T \)
11 \( 1 + (0.775 + 0.631i)T \)
13 \( 1 + (-0.334 + 0.942i)T \)
17 \( 1 + (-0.962 + 0.269i)T \)
19 \( 1 + (0.854 - 0.519i)T \)
23 \( 1 + (0.854 + 0.519i)T \)
29 \( 1 + (-0.917 + 0.398i)T \)
31 \( 1 + (-0.854 - 0.519i)T \)
37 \( 1 + (0.917 + 0.398i)T \)
41 \( 1 + (0.962 - 0.269i)T \)
43 \( 1 + (-0.203 + 0.979i)T \)
47 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (-0.460 + 0.887i)T \)
59 \( 1 + (-0.775 + 0.631i)T \)
61 \( 1 + (0.775 - 0.631i)T \)
67 \( 1 + (0.682 + 0.730i)T \)
71 \( 1 + (-0.576 + 0.816i)T \)
73 \( 1 + (-0.682 + 0.730i)T \)
79 \( 1 + (0.854 + 0.519i)T \)
83 \( 1 + (0.682 + 0.730i)T \)
89 \( 1 + (-0.990 - 0.136i)T \)
97 \( 1 + (0.576 - 0.816i)T \)
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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

−25.13220952600411892686521544773, −24.367204047967160264862864142012, −23.36059079431668153666981132501, −22.33084694286488264326390953826, −22.10662009313025991816023082044, −20.56793023854200995265888364871, −19.736518228492717509290789883625, −18.89088219072069835909293777774, −18.21287183072293112581804023492, −17.499564853982222402604870421491, −16.38316406872500158342987289690, −14.90201915066567857828365731214, −14.01269761594740467430201454860, −12.817544015315231025616732310692, −11.989870703283932166634350711827, −11.24512142350613444632685291970, −10.766644168139783284225748037742, −9.07003854884324635080161165371, −8.07053238103449643776682192311, −7.29355908960475268759264115875, −5.75120954096162472496367122105, −4.63438363582923658952443337197, −3.25424724693018232192596391017, −2.108367380849043655701865474134, −0.70764094878285538848861335593, 1.16168976343999659852637837697, 3.99051281243213022299955647439, 4.45810017246741015463674651307, 5.349443524717340369331359575240, 6.82407991088048362238368107660, 7.57887622211009946717842282421, 8.98092053421963451759490096622, 9.44233866433084649494489122601, 10.99310546309813018793632521177, 11.63466410541850347026476767704, 12.97302487263642639828283082876, 14.3697365054300946428989167907, 15.037205661719805158142615643443, 15.86849680642989709704433434147, 16.87128330003841459930434258057, 17.23666197560929272531490644376, 18.3173354908358634879161532088, 19.69895059537419040700627129359, 20.41519471519554036753948770363, 21.72320529533416034460463060241, 22.51565770245357530813264347266, 23.51897067930821661119061111335, 23.99429721487786271183654614070, 24.88355608545939067718641748960, 26.378135117134140714569054298673

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