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Properties

Label 1-277-277.146-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.447 + 0.894i$
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  + (−0.682 + 0.730i)2-s + (−0.775 − 0.631i)3-s + (−0.0682 − 0.997i)4-s + (0.576 + 0.816i)5-s + (0.990 − 0.136i)6-s + (−0.576 + 0.816i)7-s + (0.775 + 0.631i)8-s + (0.203 + 0.979i)9-s + (−0.990 − 0.136i)10-s + (0.917 + 0.398i)11-s + (−0.576 + 0.816i)12-s + (0.203 − 0.979i)13-s + (−0.203 − 0.979i)14-s + (0.0682 − 0.997i)15-s + (−0.990 + 0.136i)16-s + (−0.460 − 0.887i)17-s + ⋯
L(s)  = 1  + (−0.682 + 0.730i)2-s + (−0.775 − 0.631i)3-s + (−0.0682 − 0.997i)4-s + (0.576 + 0.816i)5-s + (0.990 − 0.136i)6-s + (−0.576 + 0.816i)7-s + (0.775 + 0.631i)8-s + (0.203 + 0.979i)9-s + (−0.990 − 0.136i)10-s + (0.917 + 0.398i)11-s + (−0.576 + 0.816i)12-s + (0.203 − 0.979i)13-s + (−0.203 − 0.979i)14-s + (0.0682 − 0.997i)15-s + (−0.990 + 0.136i)16-s + (−0.460 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3031241892 + 0.4909203430i\)
\(L(\frac12)\) \(\approx\) \(0.3031241892 + 0.4909203430i\)
\(L(1)\) \(\approx\) \(0.5498436486 + 0.2637799521i\)
\(L(1)\) \(\approx\) \(0.5498436486 + 0.2637799521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.682 + 0.730i)T \)
3 \( 1 + (-0.775 - 0.631i)T \)
5 \( 1 + (0.576 + 0.816i)T \)
7 \( 1 + (-0.576 + 0.816i)T \)
11 \( 1 + (0.917 + 0.398i)T \)
13 \( 1 + (0.203 - 0.979i)T \)
17 \( 1 + (-0.460 - 0.887i)T \)
19 \( 1 + (-0.576 + 0.816i)T \)
23 \( 1 + (-0.576 - 0.816i)T \)
29 \( 1 + (-0.0682 + 0.997i)T \)
31 \( 1 + (0.576 + 0.816i)T \)
37 \( 1 + (0.0682 + 0.997i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (-0.682 + 0.730i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (0.334 + 0.942i)T \)
59 \( 1 + (-0.917 + 0.398i)T \)
61 \( 1 + (0.917 - 0.398i)T \)
67 \( 1 + (-0.990 + 0.136i)T \)
71 \( 1 + (-0.775 - 0.631i)T \)
73 \( 1 + (0.990 + 0.136i)T \)
79 \( 1 + (-0.576 - 0.816i)T \)
83 \( 1 + (-0.990 + 0.136i)T \)
89 \( 1 + (0.854 - 0.519i)T \)
97 \( 1 + (0.775 + 0.631i)T \)
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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

−25.81392325858251065027092974306, −24.41503656593157891275818774514, −23.47613028498976593660767173215, −22.263455778318581221016526370717, −21.619429234025224999388625207764, −20.907895477227390152867379724919, −19.88069707803848667455118941942, −19.177932197438664570489710264267, −17.68435536418358763362706462095, −17.07916957733827627683271294436, −16.61729196092625038003591352774, −15.63421830699335902936354518664, −13.84260841859784254465058406168, −13.02762649079399457410455496581, −11.94707787163182532455908028430, −11.14978460608483450248042457690, −10.12034353815915376544913143712, −9.385964353718226771215418872659, −8.6334345811910213233551910351, −6.92408047438451331631265690834, −5.9922879019154370155578845916, −4.28569806338784043600426458101, −3.87462191065241014722456532397, −1.88599451875544813172236575478, −0.57273248228635685205822300757, 1.44587915954352742254547698291, 2.706497304978931461844332768139, 4.89895922560103289132182279637, 6.15948274499773326260412448655, 6.38299137397727983386611361184, 7.49085352501789383489352645060, 8.72264395920851836322935393214, 9.885442498737406616007724339351, 10.63146430920579809071896865446, 11.78127584824346546777434096438, 12.886907212571031461372650203969, 14.040635701815601028774159332, 14.94477014773621634781847462615, 15.98780980553779592675215513194, 16.897478261021878684202634508853, 17.90222350758454817731271758693, 18.30132622982282484574927828424, 19.129433608814401447563356082, 20.13120396008332317998408820819, 21.871406368053880762707915127684, 22.645796760875690658572710168410, 23.06274565086724288155735174753, 24.49957770160520893599419961029, 25.17621930521250016396178174961, 25.55015800092698964888539003326

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