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Properties

Label 2-2667-2667.1172-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.943 - 0.330i$
Analytic cond. $1.33100$
Root an. cond. $1.15369$
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  + (−0.980 − 0.198i)3-s + (0.826 − 0.563i)4-s + (−0.853 + 0.521i)7-s + (0.921 + 0.388i)9-s + (−0.921 + 0.388i)12-s + (−1.21 + 0.875i)13-s + (0.365 − 0.930i)16-s + (1.45 + 0.840i)19-s + (0.939 − 0.342i)21-s + (0.623 − 0.781i)25-s + (−0.826 − 0.563i)27-s + (−0.411 + 0.911i)28-s + (0.493 + 1.93i)31-s + (0.980 − 0.198i)36-s + (0.894 + 0.750i)37-s + ⋯
L(s)  = 1  + (−0.980 − 0.198i)3-s + (0.826 − 0.563i)4-s + (−0.853 + 0.521i)7-s + (0.921 + 0.388i)9-s + (−0.921 + 0.388i)12-s + (−1.21 + 0.875i)13-s + (0.365 − 0.930i)16-s + (1.45 + 0.840i)19-s + (0.939 − 0.342i)21-s + (0.623 − 0.781i)25-s + (−0.826 − 0.563i)27-s + (−0.411 + 0.911i)28-s + (0.493 + 1.93i)31-s + (0.980 − 0.198i)36-s + (0.894 + 0.750i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.943 - 0.330i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.943 - 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9391723277\)
\(L(\frac12)\) \(\approx\) \(0.9391723277\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.980 + 0.198i)T \)
7 \( 1 + (0.853 - 0.521i)T \)
127 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.826 + 0.563i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.583 + 0.811i)T^{2} \)
13 \( 1 + (1.21 - 0.875i)T + (0.318 - 0.947i)T^{2} \)
17 \( 1 + (-0.0249 - 0.999i)T^{2} \)
19 \( 1 + (-1.45 - 0.840i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.583 + 0.811i)T^{2} \)
29 \( 1 + (0.542 - 0.840i)T^{2} \)
31 \( 1 + (-0.493 - 1.93i)T + (-0.878 + 0.478i)T^{2} \)
37 \( 1 + (-0.894 - 0.750i)T + (0.173 + 0.984i)T^{2} \)
41 \( 1 + (-0.0249 - 0.999i)T^{2} \)
43 \( 1 + (0.307 - 0.503i)T + (-0.456 - 0.889i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.698 + 0.715i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-1.11 + 0.892i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (-0.963 - 0.850i)T + (0.124 + 0.992i)T^{2} \)
71 \( 1 + (0.583 - 0.811i)T^{2} \)
73 \( 1 + (-0.757 + 0.172i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (1.87 + 0.0936i)T + (0.995 + 0.0995i)T^{2} \)
83 \( 1 + (-0.698 - 0.715i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
97 \( 1 + (0.0182 + 0.730i)T + (-0.998 + 0.0498i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.419459289189712584618426039447, −8.176605748918504383517005424463, −7.12914887031124620512374442781, −6.79551039976568733348179122161, −6.07668293955256834897272955722, −5.31191149644984327956147739192, −4.67831085985430386999047489147, −3.24041062055138712818076210201, −2.30487017935232409621529186350, −1.17671582779674010641545177874, 0.78137105624400628338287704326, 2.47591501227508837071439521676, 3.29657869602834152423200604390, 4.21491344666635378394649095547, 5.25189812707752583995331148177, 5.91685714261651490973806795153, 6.85641908624548906589297243166, 7.29893661087061375056859677591, 7.86639874726616225475315479923, 9.276642667818741517549121318131

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