↑

Properties

Label 2-546-273.251-c1-0-15
Degree $2$
Conductor $546$
Sign $-0.430 - 0.902i$
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 + 3.46i·5-s + (−1.5 + 0.866i)6-s + (2.5 + 0.866i)7-s + 0.999·8-s + (1.5 + 2.59i)9-s + (−2.99 − 1.73i)10-s + (1.5 − 2.59i)11-s − 1.73i·12-s + (3.5 − 0.866i)13-s + (−2 + 1.73i)14-s + (−2.99 + 5.19i)15-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s + 1.54i·5-s + (−0.612 + 0.353i)6-s + (0.944 + 0.327i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (−0.948 − 0.547i)10-s + (0.452 − 0.783i)11-s − 0.499i·12-s + (0.970 − 0.240i)13-s + (−0.534 + 0.462i)14-s + (−0.774 + 1.34i)15-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.430 - 0.902i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.430 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953310 + 1.51166i\)
\(L(\frac12)\) \(\approx\) \(0.953310 + 1.51166i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6 + 3.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.66iT - 47T^{2} \)
53 \( 1 + 8.66iT - 53T^{2} \)
59 \( 1 + (9 - 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-7.5 - 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.76404171652445162436368700314, −10.35755586450647557013418473890, −9.035291318085816790716224679813, −8.445119464798784168124227129641, −7.66825729244884065434916349889, −6.63660508751633759085438156484, −5.78103953968848611458012491794, −4.33438896394422983349627880181, −3.29562297744852499127160483122, −2.05434470899961253077337944268, 1.27799469535505028144983064005, 1.87452374099748778201388988223, 3.83325223241200571344385514153, 4.42096559079614079033351548551, 5.81676974811283150742554884998, 7.39024815444517330674212964731, 8.101814200092245320152805719619, 8.739937618297979873078353338759, 9.425874682483268862394938997865, 10.34770923065753354751382546040

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