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Properties

Label 1-2667-2667.1994-r1-0-0
Degree $1$
Conductor $2667$
Sign $0.852 - 0.522i$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.222 + 0.974i)5-s + (0.900 − 0.433i)8-s + (0.623 − 0.781i)10-s + (−0.365 − 0.930i)11-s + (0.733 − 0.680i)13-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s − 19-s − 20-s + (−0.5 + 0.866i)22-s + (0.365 − 0.930i)23-s + (−0.900 + 0.433i)25-s + (−0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.222 + 0.974i)5-s + (0.900 − 0.433i)8-s + (0.623 − 0.781i)10-s + (−0.365 − 0.930i)11-s + (0.733 − 0.680i)13-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s − 19-s − 20-s + (−0.5 + 0.866i)22-s + (0.365 − 0.930i)23-s + (−0.900 + 0.433i)25-s + (−0.988 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(286.608\)
Root analytic conductor: \(286.608\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1994, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (1:\ ),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.412092612 - 0.3978401068i\)
\(L(\frac12)\) \(\approx\) \(1.412092612 - 0.3978401068i\)
\(L(1)\) \(\approx\) \(0.7988063183 - 0.1674884744i\)
\(L(1)\) \(\approx\) \(0.7988063183 - 0.1674884744i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (0.733 - 0.680i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 - T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + (-0.955 + 0.294i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 + (-0.0747 - 0.997i)T \)
71 \( 1 + (-0.365 + 0.930i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (0.365 - 0.930i)T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (0.826 + 0.563i)T \)
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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

−19.208147892881763349303456057563, −18.26544274311913006092657865527, −17.73926914101499703626569515105, −17.02213570370166031294039544642, −16.39390505971291357227306277515, −15.8889041118564735949939387237, −15.08386856748561094120305570669, −14.359402420088593279298242777140, −13.586025523933075370264806821637, −12.89199911024749628714755845023, −12.12537909891994054110310816136, −11.15778078041506275294337902232, −10.30122649117035929793977010210, −9.62346603797290068819496898459, −8.9595438548672542871034550156, −8.39309947897848067512492990569, −7.5143417884584726254085352083, −6.88523249493196053922301564111, −5.90187890414836426074470014093, −5.277099969535842197166086555799, −4.57914728971449231027767810586, −3.70162779036601450174446607580, −2.09754505323841607620049450157, −1.52482469973560850487103691735, −0.54573349548093421559592305792, 0.51886199544102728386926055423, 1.453613352116343844883457235611, 2.51378705439708095052167341155, 3.14799162546661818256843252398, 3.72162555760799361410208269938, 4.85900394236568051686402672812, 6.01951318733822349886601118488, 6.559761522812624961847915279482, 7.68519426737691351794666960542, 8.21744305355380281838754661695, 8.906267420950257052362928580482, 9.98817243228910909817757347796, 10.46405396131987320085567800832, 11.00947326381567088988976582282, 11.63715139161075427399056277480, 12.72873262631076153869079466277, 13.13183763102657393662568362440, 14.08153571427812824454308314938, 14.673969842812585743626674338614, 15.71334695544451832439103356926, 16.35562913414207646340668278899, 17.200574289179676418664680152147, 17.81097683823709169323182464995, 18.60565839961303995673675086023, 18.9264210316852269228190524229

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