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Properties

Label 1-2667-2667.1634-r1-0-0
Degree $1$
Conductor $2667$
Sign $0.992 + 0.126i$
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 + (0.900 + 0.433i)5-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)10-s + (−0.456 − 0.889i)11-s + (0.583 − 0.811i)13-s + (0.365 + 0.930i)16-s + (0.698 + 0.715i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (0.173 + 0.984i)22-s + (0.456 − 0.889i)23-s + (0.623 + 0.781i)25-s + (−0.797 + 0.603i)26-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (0.900 + 0.433i)5-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)10-s + (−0.456 − 0.889i)11-s + (0.583 − 0.811i)13-s + (0.365 + 0.930i)16-s + (0.698 + 0.715i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (0.173 + 0.984i)22-s + (0.456 − 0.889i)23-s + (0.623 + 0.781i)25-s + (−0.797 + 0.603i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.992 + 0.126i$
Analytic conductor: \(286.608\)
Root analytic conductor: \(286.608\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1634, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (1:\ ),\ 0.992 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.249672840 + 0.1423319181i\)
\(L(\frac12)\) \(\approx\) \(2.249672840 + 0.1423319181i\)
\(L(1)\) \(\approx\) \(0.9917008767 - 0.04233853394i\)
\(L(1)\) \(\approx\) \(0.9917008767 - 0.04233853394i\)

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 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (-0.456 - 0.889i)T \)
13 \( 1 + (0.583 - 0.811i)T \)
17 \( 1 + (0.698 + 0.715i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.456 - 0.889i)T \)
29 \( 1 + (0.878 + 0.478i)T \)
31 \( 1 + (0.969 + 0.246i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (0.698 + 0.715i)T \)
43 \( 1 + (0.853 - 0.521i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.921 - 0.388i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 + (0.661 + 0.749i)T \)
71 \( 1 + (-0.456 + 0.889i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + (-0.998 + 0.0498i)T \)
83 \( 1 + (-0.921 + 0.388i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
97 \( 1 + (-0.0249 + 0.999i)T \)
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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

−18.968981694494617653496468350060, −18.26060395538420416042647043545, −17.68042043265392696319643078116, −17.18613028043895174286196541717, −16.35634736715756415503877606618, −15.79177601983933274630617733096, −15.09534000626976318527256370469, −14.083961214746657912819997679001, −13.626664914400766094190827689768, −12.65714524617873207247025050753, −11.76562851781006277044261361999, −11.17194630187128306807664195663, −10.10923739639756162216419917775, −9.67732492844033734214651149122, −9.12670193988560287040531897555, −8.29095871067694382635750286646, −7.44614988107434992505257477742, −6.76571065746377814716714991505, −5.98819908451533355255942597984, −5.19130399986236967600459707769, −4.501607796647129507433760136376, −2.95702347290640440323382437480, −2.28025021199216712737415135349, −1.348153776608956860824290033425, −0.67417713830595740658162895601, 0.8272433759227997900867671845, 1.31306163628159685271679861495, 2.60602778441575892813427300105, 2.979795767277674969580942125626, 3.93281887027262396310351845718, 5.43848223143098727800849708167, 5.9954106425578954091134929900, 6.66505743858815731741873900339, 7.72115806549373872498195351514, 8.306968774217415705759695614625, 8.977511612053692129353471301577, 9.989085762534383349284498163141, 10.41134615700527827270099931157, 10.91763661970599227997911432428, 11.844679795230108100284857884865, 12.756870365631459819382975952521, 13.26004950709778685755409859458, 14.29371746745778769404314599130, 14.86136279623106043956247576281, 16.00100034693529596803870528206, 16.334488754287254903035584924523, 17.23702087968748734002960363207, 17.844719465157942481065085307511, 18.44534330666857189861461955733, 18.952525131837747956376658723934

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