↑

Properties

Label 2-546-273.173-c1-0-18
Degree $2$
Conductor $546$
Sign $-0.313 - 0.949i$
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  + (0.5 + 0.866i)2-s + (1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 0.866i)5-s + 1.73i·6-s + (0.5 + 2.59i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + 1.73i·10-s − 3·11-s + (−1.49 + 0.866i)12-s + (1 − 3.46i)13-s + (−2 + 1.73i)14-s + (1.5 + 2.59i)15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.670 + 0.387i)5-s + 0.707i·6-s + (0.188 + 0.981i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + 0.547i·10-s − 0.904·11-s + (−0.433 + 0.250i)12-s + (0.277 − 0.960i)13-s + (−0.534 + 0.462i)14-s + (0.387 + 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41417 + 1.95590i\)
\(L(\frac12)\) \(\approx\) \(1.41417 + 1.95590i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
13 \( 1 + (-1 + 3.46i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.5 - 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.5 + 0.866i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 0.866i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9 - 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.73iT - 61T^{2} \)
67 \( 1 + 1.73iT - 67T^{2} \)
71 \( 1 + (4.5 + 7.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.74372793858165603646458344877, −10.15586007818550702984468876163, −9.107092658571678320926294911471, −8.398047365693594049626957191404, −7.64584527518453847966192925392, −6.41281119721900638368003076189, −5.41539999020835388894773984016, −4.67657396129137277749226430290, −3.05817281814675295034763128246, −2.45498288846257318446906397101, 1.31280746891108112408432581906, 2.30842711932641985125490006132, 3.69958905311988802521932387899, 4.54881747004246941964133027837, 5.94003893984480475376863647178, 6.88711993768394632597413624108, 8.058060164040389894862853771617, 8.739548404838418587888437191782, 9.834938720355728077435155031090, 10.40599021814903233494159102102

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