Properties

Label 2-546-91.16-c1-0-12
Degree $2$
Conductor $546$
Sign $0.770 + 0.637i$
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 + (−0.5 + 0.866i)3-s + 4-s + (1.75 − 3.03i)5-s + (−0.5 + 0.866i)6-s + (−2.63 + 0.222i)7-s + 8-s + (−0.499 − 0.866i)9-s + (1.75 − 3.03i)10-s + (3.20 − 5.54i)11-s + (−0.5 + 0.866i)12-s + (−0.213 + 3.59i)13-s + (−2.63 + 0.222i)14-s + (1.75 + 3.03i)15-s + 16-s + 4.67·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (0.784 − 1.35i)5-s + (−0.204 + 0.353i)6-s + (−0.996 + 0.0839i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.554 − 0.960i)10-s + (0.966 − 1.67i)11-s + (−0.144 + 0.249i)12-s + (−0.0590 + 0.998i)13-s + (−0.704 + 0.0593i)14-s + (0.452 + 0.784i)15-s + 0.250·16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00881 - 0.722694i\)
\(L(\frac12)\) \(\approx\) \(2.00881 - 0.722694i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.63 - 0.222i)T \)
13 \( 1 + (0.213 - 3.59i)T \)
good5 \( 1 + (-1.75 + 3.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.20 + 5.54i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.67T + 17T^{2} \)
19 \( 1 + (2.61 + 4.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.16T + 23T^{2} \)
29 \( 1 + (1.23 + 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.46 - 7.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.89T + 37T^{2} \)
41 \( 1 + (-5.09 - 8.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.19 - 2.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.44 - 4.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.05 + 1.82i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + (4.67 + 8.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.64 - 6.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.79 + 4.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.23 - 7.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.893 + 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.59T + 83T^{2} \)
89 \( 1 - 7.00T + 89T^{2} \)
97 \( 1 + (4.92 - 8.53i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.82471077461316841049547381545, −9.576474063653785975394943592798, −9.226175695029318103997723912879, −8.261557847957081455755408274663, −6.42327046264987294231018976348, −6.12335429545705566117955748740, −5.06539567021044831000545993814, −4.13780680459462548692319863473, −3.01428667893019982989718494949, −1.13218417911323628651200122569, 1.91268365663615367596141253890, 2.98498107042203867014697177399, 4.07881470209480537671837244214, 5.73642008675273007933450156477, 6.21742802771706712084153917499, 7.06077778543190808032088726083, 7.72054985109260349579645179533, 9.610745053895172106226141668199, 10.09722027179592433665635136268, 10.79827343002142325384580981354

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