↑

Properties

Label 2-546-91.89-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.567 - 0.823i$
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  + (0.707 + 0.707i)2-s + (0.866 + 0.5i)3-s + 1.00i·4-s + (0.349 + 1.30i)5-s + (0.258 + 0.965i)6-s + (−2.23 + 1.41i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.676 + 1.17i)10-s + (0.147 + 0.549i)11-s + (−0.500 + 0.866i)12-s + (−3.33 + 1.36i)13-s + (−2.58 − 0.579i)14-s + (−0.349 + 1.30i)15-s − 1.00·16-s + 0.975·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.499 + 0.288i)3-s + 0.500i·4-s + (0.156 + 0.584i)5-s + (0.105 + 0.394i)6-s + (−0.844 + 0.535i)7-s + (−0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (−0.213 + 0.370i)10-s + (0.0444 + 0.165i)11-s + (−0.144 + 0.250i)12-s + (−0.925 + 0.379i)13-s + (−0.689 − 0.154i)14-s + (−0.0903 + 0.337i)15-s − 0.250·16-s + 0.236·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895118 + 1.70479i\)
\(L(\frac12)\) \(\approx\) \(0.895118 + 1.70479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (2.23 - 1.41i)T \)
13 \( 1 + (3.33 - 1.36i)T \)
good5 \( 1 + (-0.349 - 1.30i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.147 - 0.549i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.975T + 17T^{2} \)
19 \( 1 + (-3.14 - 0.841i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.0810iT - 23T^{2} \)
29 \( 1 + (0.139 + 0.242i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.48 - 0.665i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.14 + 1.14i)T - 37iT^{2} \)
41 \( 1 + (-2.23 - 0.599i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.48 - 5.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.55 - 1.48i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.54 + 7.86i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.05 - 5.05i)T + 59iT^{2} \)
61 \( 1 + (-4.95 + 2.86i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.09 + 0.561i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.466 + 0.124i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.753 - 2.81i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.45 + 2.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.22 - 2.22i)T - 83iT^{2} \)
89 \( 1 + (7.48 + 7.48i)T + 89iT^{2} \)
97 \( 1 + (-4.34 - 16.2i)T + (-84.0 + 48.5i)T^{2} \)
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

−11.14539729347704578017564547950, −9.919887282350597728357857521990, −9.482964303222646251714537608450, −8.377625663514132815928472002216, −7.34865909582657018881083425930, −6.59944009918961645056885789566, −5.60253122813751360816960924044, −4.50142018169912618252449786454, −3.27430559529564577984805360083, −2.47058462193161036603706415378, 0.921866877285163965244015039788, 2.57596608882276950139980076347, 3.53883945698977501029101929682, 4.70275185972855443417181127332, 5.73757279931632485060021925301, 6.88461829134247246567344913178, 7.73099171307082548025194823914, 8.968922304180950509898723628263, 9.670049815333119958485225940852, 10.39136370569774964456398541714

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