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Properties

Label 1-2667-2667.641-r0-0-0
Degree $1$
Conductor $2667$
Sign $-0.100 + 0.994i$
Analytic cond. $12.3854$
Root an. cond. $12.3854$
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 + (−0.921 − 0.388i)11-s + (−0.969 + 0.246i)13-s + (0.365 + 0.930i)16-s + (0.411 + 0.911i)17-s + 19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.921 − 0.388i)23-s + (−0.988 + 0.149i)25-s + (0.998 + 0.0498i)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 + (−0.921 − 0.388i)11-s + (−0.969 + 0.246i)13-s + (0.365 + 0.930i)16-s + (0.411 + 0.911i)17-s + 19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.921 − 0.388i)23-s + (−0.988 + 0.149i)25-s + (0.998 + 0.0498i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ -0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5475674144 + 0.6056441169i\)
\(L(\frac12)\) \(\approx\) \(0.5475674144 + 0.6056441169i\)
\(L(1)\) \(\approx\) \(0.6646349274 + 0.1291269679i\)
\(L(1)\) \(\approx\) \(0.6646349274 + 0.1291269679i\)

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 \( 1 + (-0.921 - 0.388i)T \)
13 \( 1 + (-0.969 + 0.246i)T \)
17 \( 1 + (0.411 + 0.911i)T \)
19 \( 1 + T \)
23 \( 1 + (0.921 - 0.388i)T \)
29 \( 1 + (0.980 - 0.198i)T \)
31 \( 1 + (-0.411 + 0.911i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.583 - 0.811i)T \)
43 \( 1 + (0.661 + 0.749i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (0.542 + 0.840i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.878 - 0.478i)T \)
71 \( 1 + (0.124 - 0.992i)T \)
73 \( 1 + (-0.733 - 0.680i)T \)
79 \( 1 + (-0.124 + 0.992i)T \)
83 \( 1 + (0.456 - 0.889i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.318 + 0.947i)T \)
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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

−19.166696728589001563146591882936, −18.19137433888663494058637163767, −17.76428064910995857181087720595, −17.05273084000527073673574323976, −16.25143138525273166282596732959, −15.93758984812912284022595988364, −15.06011059195310959392277036855, −14.3431414009054151066426369045, −13.35435518297048425801039450304, −12.61925806081949330050999720346, −11.87168194050843882903356413896, −11.20618737811515316578320524023, −10.1560017937980921188314629337, −9.61869133616714565541133877745, −9.128177268590434437448050714088, −8.09703219820775760181098218243, −7.59588712438540232046514079835, −6.99171283157574824324811688067, −5.6899591088832535991216168445, −5.26497408597509291946310513534, −4.52674341643975953546619584023, −3.00186725686157251203210562730, −2.36140318091180040629128841238, −1.24304807451746431041466052825, −0.43457361318227949056095288029, 0.97928818176434272001910208892, 2.09697878321768408028151835356, 2.891468793817382383809174533259, 3.31755375074688707842217190607, 4.607991375391977853750097198312, 5.719279226521659502834602340773, 6.470421219479518866501115402632, 7.36108791004930228229533713084, 7.7186710616564806399295798644, 8.66181193585690519743257795364, 9.49591019496857305927796135511, 10.32531600295879665497798021802, 10.58641033494994976731526808269, 11.465985706179169518923669269676, 12.157560153164925701490526718861, 12.929015713693234661625198603865, 13.86625099636374135654429812481, 14.72271937979698513457104074200, 15.2992766281529031421181154107, 16.12230774672143329561124596411, 16.77187992698744170153583795019, 17.6051131364978971025696349116, 18.14941385013579486251944822065, 18.75422313009589258222674814099, 19.43670669275562194571786780152

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