↑

Properties

Label 2-546-91.20-c1-0-4
Degree $2$
Conductor $546$
Sign $0.934 + 0.356i$
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 + (−0.866 − 0.5i)3-s + (0.866 − 0.499i)4-s + (−1.23 + 1.23i)5-s + (0.965 + 0.258i)6-s + (−2.50 + 0.855i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.870 − 1.50i)10-s + (−1.51 − 5.65i)11-s − 12-s + (3.23 + 1.59i)13-s + (2.19 − 1.47i)14-s + (1.68 − 0.450i)15-s + (0.500 − 0.866i)16-s + (0.548 + 0.950i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.550 + 0.550i)5-s + (0.394 + 0.105i)6-s + (−0.946 + 0.323i)7-s + (−0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (0.275 − 0.477i)10-s + (−0.456 − 1.70i)11-s − 0.288·12-s + (0.897 + 0.441i)13-s + (0.587 − 0.393i)14-s + (0.434 − 0.116i)15-s + (0.125 − 0.216i)16-s + (0.133 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657641 - 0.121250i\)
\(L(\frac12)\) \(\approx\) \(0.657641 - 0.121250i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (2.50 - 0.855i)T \)
13 \( 1 + (-3.23 - 1.59i)T \)
good5 \( 1 + (1.23 - 1.23i)T - 5iT^{2} \)
11 \( 1 + (1.51 + 5.65i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.548 - 0.950i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.79 - 1.01i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.260 - 0.150i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.99 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.12 + 6.12i)T - 31iT^{2} \)
37 \( 1 + (-0.247 - 0.924i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.105 + 0.392i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.81 + 5.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.94 + 3.94i)T + 47iT^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + (3.62 - 13.5i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.572 - 0.330i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.17 + 1.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.02 + 11.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.47 + 1.47i)T + 73iT^{2} \)
79 \( 1 + 1.34T + 79T^{2} \)
83 \( 1 + (3.55 - 3.55i)T - 83iT^{2} \)
89 \( 1 + (0.614 - 0.164i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (15.9 + 4.26i)T + (84.0 + 48.5i)T^{2} \)
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

−10.79531586524429957076578948552, −9.945261930986159863697882246486, −8.874668071473427597722376828860, −8.091791758921057396728363082864, −7.15924509925989816667247684826, −6.14448933076530555893473297516, −5.72632551797914893342584375306, −3.78241592588098465313915617398, −2.76625021380112476538059742374, −0.70832444552025619460960505110, 0.976347964329616113998807977575, 2.92666601101017648901318876770, 4.16930311837267851260680280404, 5.17474192010387073604196886789, 6.52068766596371797480240024092, 7.28668533809501911134341430621, 8.245218750178874368166929127151, 9.288023970450081752318642256001, 10.01394764619408554701387355910, 10.61578934926940236774476195321

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