↑

Properties

Label 2-2667-2667.11-c0-0-0
Degree $2$
Conductor $2667$
Sign $-0.0383 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.797 − 0.603i)3-s + (0.365 + 0.930i)4-s + (0.542 − 0.840i)7-s + (0.270 + 0.962i)9-s + (0.270 − 0.962i)12-s + (−1.21 + 1.38i)13-s + (−0.733 + 0.680i)16-s + (−0.995 + 1.72i)19-s + (−0.939 + 0.342i)21-s + (−0.222 + 0.974i)25-s + (0.365 − 0.930i)27-s + (0.980 + 0.198i)28-s + (−0.0227 − 0.912i)31-s + (−0.797 + 0.603i)36-s + (−1.01 − 0.850i)37-s + ⋯
L(s)  = 1  + (−0.797 − 0.603i)3-s + (0.365 + 0.930i)4-s + (0.542 − 0.840i)7-s + (0.270 + 0.962i)9-s + (0.270 − 0.962i)12-s + (−1.21 + 1.38i)13-s + (−0.733 + 0.680i)16-s + (−0.995 + 1.72i)19-s + (−0.939 + 0.342i)21-s + (−0.222 + 0.974i)25-s + (0.365 − 0.930i)27-s + (0.980 + 0.198i)28-s + (−0.0227 − 0.912i)31-s + (−0.797 + 0.603i)36-s + (−1.01 − 0.850i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0383 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0383 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.0383 - 0.999i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ -0.0383 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7484596177\)
\(L(\frac12)\) \(\approx\) \(0.7484596177\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.797 + 0.603i)T \)
7 \( 1 + (-0.542 + 0.840i)T \)
127 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.365 - 0.930i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.661 - 0.749i)T^{2} \)
13 \( 1 + (1.21 - 1.38i)T + (-0.124 - 0.992i)T^{2} \)
17 \( 1 + (-0.456 - 0.889i)T^{2} \)
19 \( 1 + (0.995 - 1.72i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.661 + 0.749i)T^{2} \)
29 \( 1 + (-0.995 + 0.0995i)T^{2} \)
31 \( 1 + (0.0227 + 0.912i)T + (-0.998 + 0.0498i)T^{2} \)
37 \( 1 + (1.01 + 0.850i)T + (0.173 + 0.984i)T^{2} \)
41 \( 1 + (-0.456 - 0.889i)T^{2} \)
43 \( 1 + (-0.896 - 1.38i)T + (-0.411 + 0.911i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.853 - 0.521i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.379 - 1.66i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 + (-1.41 + 0.595i)T + (0.698 - 0.715i)T^{2} \)
71 \( 1 + (0.661 + 0.749i)T^{2} \)
73 \( 1 + (0.488 + 0.235i)T + (0.623 + 0.781i)T^{2} \)
79 \( 1 + (-1.09 - 1.52i)T + (-0.318 + 0.947i)T^{2} \)
83 \( 1 + (0.853 - 0.521i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
97 \( 1 + (0.668 + 1.30i)T + (-0.583 + 0.811i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.173937785631267965411429112476, −8.084023132016507417046525512798, −7.64576913267493815193795758359, −7.02142033170025172409977614475, −6.40537668204934321446825992946, −5.39148150185637902353981232187, −4.34926727395288816640476854713, −3.89416781226884900896419502451, −2.32400708880419185955042505508, −1.63426173624828520785368498218, 0.50739429815484118815325650830, 2.10669945101968889902128303652, 2.95153719480309601719066963058, 4.47651900760333320588998688217, 5.14673080003687182783915245801, 5.47149124954072533454347064975, 6.47733652150497554854150904708, 7.02407158268005972320135485469, 8.195920012435395120289203718486, 9.021442097347866963853866016902

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