Properties

Label 2-546-273.173-c1-0-19
Degree $2$
Conductor $546$
Sign $0.999 + 0.0158i$
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 + (1.71 − 0.243i)3-s + (−0.499 + 0.866i)4-s + (2.87 + 1.66i)5-s + (−1.06 − 1.36i)6-s + (−0.187 + 2.63i)7-s + 0.999·8-s + (2.88 − 0.834i)9-s − 3.32i·10-s − 1.48·11-s + (−0.646 + 1.60i)12-s + (−1.88 − 3.07i)13-s + (2.37 − 1.15i)14-s + (5.34 + 2.15i)15-s + (−0.5 − 0.866i)16-s + (−2.81 + 4.88i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.990 − 0.140i)3-s + (−0.249 + 0.433i)4-s + (1.28 + 0.743i)5-s + (−0.436 − 0.556i)6-s + (−0.0707 + 0.997i)7-s + 0.353·8-s + (0.960 − 0.278i)9-s − 1.05i·10-s − 0.447·11-s + (−0.186 + 0.463i)12-s + (−0.523 − 0.851i)13-s + (0.635 − 0.309i)14-s + (1.37 + 0.555i)15-s + (−0.125 − 0.216i)16-s + (−0.683 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0158i$
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: \((2,\ 546,\ (\ :1/2),\ 0.999 + 0.0158i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96035 - 0.0155191i\)
\(L(\frac12)\) \(\approx\) \(1.96035 - 0.0155191i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.243i)T \)
7 \( 1 + (0.187 - 2.63i)T \)
13 \( 1 + (1.88 + 3.07i)T \)
good5 \( 1 + (-2.87 - 1.66i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.48T + 11T^{2} \)
17 \( 1 + (2.81 - 4.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 5.36T + 19T^{2} \)
23 \( 1 + (3.47 - 2.00i)T + (11.5 - 19.9i)T^{2} \)
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 \( 1 + (-8.80 + 5.08i)T + (18.5 - 32.0i)T^{2} \)
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 \( 1 + (0.131 + 0.0757i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.30iT - 61T^{2} \)
67 \( 1 + 9.94iT - 67T^{2} \)
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 \( 1 + (-1.49 + 0.863i)T + (44.5 - 77.0i)T^{2} \)
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

−10.39090830038210157980118823348, −9.948525421926989188233363507186, −9.210116843548647888036623506085, −8.330289871136308559575532369334, −7.44000676420160450528887232166, −6.23279847257289965781382761632, −5.27494087292972020151753726119, −3.53954539634336364449507745733, −2.54737842124222657125950652359, −1.93105711966556339950242863231, 1.33532169333596677403361886210, 2.67105378453262013652431285474, 4.40105130308096651742102427964, 5.07897828693442288433375015703, 6.45298621091037780229412741885, 7.31301006972220205093251110613, 8.152061485035593830782292172594, 9.234351992782190385161530882744, 9.658346534265922889599911285356, 10.22202758567152170030018687132

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