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Properties

Label 1-2667-2667.824-r0-0-0
Degree $1$
Conductor $2667$
Sign $-0.810 - 0.585i$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.623 + 0.781i)5-s + (0.222 − 0.974i)8-s + (−0.0747 − 0.997i)10-s + (−0.995 − 0.0995i)11-s + (−0.980 − 0.198i)13-s + (−0.733 + 0.680i)16-s + (0.878 − 0.478i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (−0.995 + 0.0995i)23-s + (−0.222 + 0.974i)25-s + (0.698 + 0.715i)26-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.623 + 0.781i)5-s + (0.222 − 0.974i)8-s + (−0.0747 − 0.997i)10-s + (−0.995 − 0.0995i)11-s + (−0.980 − 0.198i)13-s + (−0.733 + 0.680i)16-s + (0.878 − 0.478i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (−0.995 + 0.0995i)23-s + (−0.222 + 0.974i)25-s + (0.698 + 0.715i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.810 - 0.585i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ -0.810 - 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1179861855 - 0.3648993567i\)
\(L(\frac12)\) \(\approx\) \(0.1179861855 - 0.3648993567i\)
\(L(1)\) \(\approx\) \(0.6370361986 - 0.1062632797i\)
\(L(1)\) \(\approx\) \(0.6370361986 - 0.1062632797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.995 - 0.0995i)T \)
13 \( 1 + (-0.980 - 0.198i)T \)
17 \( 1 + (0.878 - 0.478i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.995 + 0.0995i)T \)
29 \( 1 + (-0.456 + 0.889i)T \)
31 \( 1 + (0.853 - 0.521i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.878 - 0.478i)T \)
43 \( 1 + (-0.998 + 0.0498i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.969 + 0.246i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (-0.797 - 0.603i)T \)
71 \( 1 + (-0.995 + 0.0995i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (-0.411 + 0.911i)T \)
83 \( 1 + (-0.969 - 0.246i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (-0.542 + 0.840i)T \)
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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

−19.45050493494141246548037819293, −18.82053270898028907126030734416, −18.00000789466141210775562390851, −17.50132202050289934478905206140, −16.74812004602923502848442029100, −16.26782866882666442894394242298, −15.57216246695575423544103714447, −14.68125768624970030958998957102, −14.0626004693937602417429043904, −13.32384739895264603916913286469, −12.30545590591144113743531055888, −11.860046020107568242327168444756, −10.538023612080168052304166825628, −10.034448843986671430022778290074, −9.6015983621401830899096127658, −8.629217228398475351502223903643, −7.94081125763445687037566717596, −7.46482265927021036605049434227, −6.321226646549191308231725226557, −5.62416406789695584742380320760, −5.13221645543088200728818371672, −4.17061142574230609171416886374, −2.71622808595465206503181175791, −1.94682285649778600106571534513, −1.109582528327609238347759600106, 0.16339214230060440820591206714, 1.45891346261209562661817887488, 2.51525752526735498341690099063, 2.80612584284849853585171091229, 3.766242879606227956996844610190, 5.03437123923987891344569093529, 5.733066556533483739991798542, 6.92085232932593413985442567089, 7.35634358155608541779120887753, 8.09502891466930119219056679915, 9.05763452228367083488492110206, 9.9436275314199747377578031613, 10.14363240485655334885818861423, 10.96908953556182632041900064037, 11.767345376675148764473504574687, 12.405929446334872795978288168000, 13.35003124557954293397393306216, 13.871264556674559827042949317302, 14.862467595288222803275628866799, 15.60922253322918222631528423357, 16.34828269506262092866093955982, 17.18404882160738970896661609082, 17.749675178120293545316700724, 18.403202391094929363936535935614, 18.83328968785150464462659119078

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