↑

Properties

Label 2-546-39.32-c1-0-10
Degree $2$
Conductor $546$
Sign $0.958 - 0.285i$
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  + (−0.965 + 0.258i)2-s + (1.59 + 0.674i)3-s + (0.866 − 0.499i)4-s + (−1.28 − 1.28i)5-s + (−1.71 − 0.238i)6-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (2.09 + 2.15i)9-s + (1.57 + 0.907i)10-s + (−1.42 − 5.30i)11-s + (1.71 − 0.213i)12-s + (3.42 + 1.13i)13-s − i·14-s + (−1.18 − 2.91i)15-s + (0.500 − 0.866i)16-s + (3.95 + 6.85i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.921 + 0.389i)3-s + (0.433 − 0.249i)4-s + (−0.574 − 0.574i)5-s + (−0.700 − 0.0972i)6-s + (−0.0978 + 0.365i)7-s + (−0.249 + 0.249i)8-s + (0.697 + 0.716i)9-s + (0.497 + 0.287i)10-s + (−0.428 − 1.60i)11-s + (0.496 − 0.0617i)12-s + (0.949 + 0.313i)13-s − 0.267i·14-s + (−0.305 − 0.752i)15-s + (0.125 − 0.216i)16-s + (0.959 + 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.958 - 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38555 + 0.202175i\)
\(L(\frac12)\) \(\approx\) \(1.38555 + 0.202175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-1.59 - 0.674i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-3.42 - 1.13i)T \)
good5 \( 1 + (1.28 + 1.28i)T + 5iT^{2} \)
11 \( 1 + (1.42 + 5.30i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.95 - 6.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.30 - 1.68i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.33 + 4.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.994 - 0.573i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.34 + 2.34i)T - 31iT^{2} \)
37 \( 1 + (-1.82 + 0.489i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.51 - 2.28i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.49 - 3.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.34 + 7.34i)T - 47iT^{2} \)
53 \( 1 - 2.74iT - 53T^{2} \)
59 \( 1 + (5.23 + 1.40i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.94 + 6.82i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.85 - 10.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.69 - 6.31i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.12 + 2.12i)T + 73iT^{2} \)
79 \( 1 - 0.559T + 79T^{2} \)
83 \( 1 + (-11.4 - 11.4i)T + 83iT^{2} \)
89 \( 1 + (3.08 + 11.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (14.0 + 3.75i)T + (84.0 + 48.5i)T^{2} \)
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

−10.61782795210759231558611950270, −9.868166309613845991662097053636, −8.650701164175876697283201426774, −8.458652889054999133388997046417, −7.78042020148574334311501351187, −6.31322037407400924955770327567, −5.34083396332746097715078671332, −3.86396765909548030514727229640, −3.02733251287984595431824740494, −1.24655339348540148681142245992, 1.25762781858586732730345080135, 2.84341080149759920301513998503, 3.52013584673053484047630111213, 5.04741770368526314659373214859, 6.82036067494176586629887801686, 7.45928441820219399327477773724, 7.79143826372623940244691668638, 9.164691942779751485230077233970, 9.714279108387911344647207384232, 10.55222363148598041281402480788

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