Properties

Label 2-546-273.257-c1-0-7
Degree $2$
Conductor $546$
Sign $0.995 + 0.0971i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−0.646 + 1.60i)3-s + 4-s + (−2.87 − 1.66i)5-s + (0.646 − 1.60i)6-s + (−2.37 + 1.15i)7-s − 8-s + (−2.16 − 2.07i)9-s + (2.87 + 1.66i)10-s + (−0.741 + 1.28i)11-s + (−0.646 + 1.60i)12-s + (1.88 − 3.07i)13-s + (2.37 − 1.15i)14-s + (4.53 − 3.55i)15-s + 16-s + 5.63·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.373 + 0.927i)3-s + 0.5·4-s + (−1.28 − 0.743i)5-s + (0.263 − 0.655i)6-s + (−0.899 + 0.437i)7-s − 0.353·8-s + (−0.721 − 0.692i)9-s + (0.910 + 0.525i)10-s + (−0.223 + 0.387i)11-s + (−0.186 + 0.463i)12-s + (0.523 − 0.851i)13-s + (0.635 − 0.309i)14-s + (1.17 − 0.917i)15-s + 0.250·16-s + 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.995 + 0.0971i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.995 + 0.0971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543923 - 0.0264787i\)
\(L(\frac12)\) \(\approx\) \(0.543923 - 0.0264787i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.646 - 1.60i)T \)
7 \( 1 + (2.37 - 1.15i)T \)
13 \( 1 + (-1.88 + 3.07i)T \)
good5 \( 1 + (2.87 + 1.66i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.741 - 1.28i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.63T + 17T^{2} \)
19 \( 1 + (-2.68 - 4.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + (0.127 - 0.0736i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.689 - 1.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 + (0.728 - 0.420i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.56 - 7.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.41 - 4.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.6 + 6.15i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.151iT - 59T^{2} \)
61 \( 1 + (-6.32 + 3.65i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.61 + 4.97i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.25 + 3.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.99 - 3.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.75 + 3.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + 1.72iT - 89T^{2} \)
97 \( 1 + (-7.27 + 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.54330615384629843943611276776, −9.962048803255778676695806647883, −9.054145113875294148928750509132, −8.262324639946814066438200561397, −7.50564540351412784331974743778, −6.06679851696047336067573621480, −5.25988114733245956401373553497, −3.91877901673458197705365888723, −3.14319372120692648716821000445, −0.60701741131924219119454624429, 0.891791541730053080981781407807, 2.84293193946886824054092014927, 3.74528163874011356533910919917, 5.57270216458691983230535392547, 6.74843236482887171288116413635, 7.17955007310641917208598850381, 7.923353313534430272870781833206, 8.881292535557988311020255084112, 10.09495096976206070560646792510, 10.89655402844537036080124793759

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