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Properties

Label 1-2667-2667.1277-r1-0-0
Degree $1$
Conductor $2667$
Sign $-0.681 + 0.731i$
Analytic cond. $286.608$
Root an. cond. $286.608$
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  + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (−0.0747 − 0.997i)5-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)10-s + (0.124 + 0.992i)11-s + (−0.270 + 0.962i)13-s + (0.365 + 0.930i)16-s + (−0.583 + 0.811i)17-s − 19-s + (0.5 − 0.866i)20-s + (0.173 − 0.984i)22-s + (−0.124 + 0.992i)23-s + (−0.988 + 0.149i)25-s + (0.542 − 0.840i)26-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (−0.0747 − 0.997i)5-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)10-s + (0.124 + 0.992i)11-s + (−0.270 + 0.962i)13-s + (0.365 + 0.930i)16-s + (−0.583 + 0.811i)17-s − 19-s + (0.5 − 0.866i)20-s + (0.173 − 0.984i)22-s + (−0.124 + 0.992i)23-s + (−0.988 + 0.149i)25-s + (0.542 − 0.840i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.681 + 0.731i$
Analytic conductor: \(286.608\)
Root analytic conductor: \(286.608\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (1:\ ),\ -0.681 + 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2013285990 + 0.4624677666i\)
\(L(\frac12)\) \(\approx\) \(0.2013285990 + 0.4624677666i\)
\(L(1)\) \(\approx\) \(0.6236428494 + 0.02620746161i\)
\(L(1)\) \(\approx\) \(0.6236428494 + 0.02620746161i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.955 - 0.294i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (0.124 + 0.992i)T \)
13 \( 1 + (-0.270 + 0.962i)T \)
17 \( 1 + (-0.583 + 0.811i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.124 + 0.992i)T \)
29 \( 1 + (-0.661 - 0.749i)T \)
31 \( 1 + (0.583 + 0.811i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.995 + 0.0995i)T \)
43 \( 1 + (0.318 - 0.947i)T \)
47 \( 1 + (0.826 + 0.563i)T \)
53 \( 1 + (0.456 - 0.889i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.853 - 0.521i)T \)
71 \( 1 + (0.797 + 0.603i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.797 - 0.603i)T \)
83 \( 1 + (0.998 - 0.0498i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.980 + 0.198i)T \)
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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

−18.625937321303579320308000209546, −18.3985968316936632335151945678, −17.47195382293826475257520883630, −16.90989718094909949782091074923, −16.05887737059486471830938988932, −15.42388702195311368711508314147, −14.76578751583802735135195370226, −14.179016885999555067203765240949, −13.26386625068091708486830577175, −12.26313066393644632178130555786, −11.27586772167497181111606345166, −10.93364221754807394599455682834, −10.24888944938710922720303676755, −9.48223971513070666985309930974, −8.60608005129773088475051170207, −7.97820639405797987965347206856, −7.239365903683310098477222297413, −6.41885036967208318035261619416, −5.98471557211298105488082928073, −4.93952779728930307052181701831, −3.707397244407467433484186305147, −2.707571724011841240120195328884, −2.316141927469587557030753195637, −0.85995823685541192082165618559, −0.15219530566409541368054090828, 0.93921185555963407935880637746, 1.933543734392290943548038388305, 2.27039787768279881393756676471, 3.95095350802826830112084050254, 4.18589345593216667037442523427, 5.40713614169620377513983847689, 6.36880203600604339153120278926, 7.14179680645838475891037585486, 7.88196730139248257322290908250, 8.67904535652277833253354303109, 9.255733678725214861346558374902, 9.81721976453999898172650239971, 10.72079404030484129758078459434, 11.4894168613493367074785484876, 12.24068445325018545833792136025, 12.695659747971730683999328650010, 13.4801052125912022742488723788, 14.62052397863480031652713172036, 15.421308510262053262809626142827, 15.97444820388550855543358907792, 16.81622005666520071569652312877, 17.39976313810570720426543483859, 17.69209359616213182313773053900, 18.881413094796264496276631127544, 19.4760632806807989455486444969

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