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Properties

Label 2-546-273.254-c1-0-30
Degree $2$
Conductor $546$
Sign $0.159 + 0.987i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (−1.23 + 1.21i)3-s + (0.866 + 0.499i)4-s + (−0.473 − 1.76i)5-s + (−1.50 + 0.851i)6-s + (−0.491 − 2.59i)7-s + (0.707 + 0.707i)8-s + (0.0596 − 2.99i)9-s − 1.83i·10-s + (−2.98 − 2.98i)11-s + (−1.67 + 0.431i)12-s + (−3.58 − 0.380i)13-s + (0.198 − 2.63i)14-s + (2.72 + 1.61i)15-s + (0.500 + 0.866i)16-s + (0.234 − 0.406i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.714 + 0.700i)3-s + (0.433 + 0.249i)4-s + (−0.211 − 0.790i)5-s + (−0.615 + 0.347i)6-s + (−0.185 − 0.982i)7-s + (0.249 + 0.249i)8-s + (0.0198 − 0.999i)9-s − 0.578i·10-s + (−0.898 − 0.898i)11-s + (−0.484 + 0.124i)12-s + (−0.994 − 0.105i)13-s + (0.0530 − 0.705i)14-s + (0.704 + 0.416i)15-s + (0.125 + 0.216i)16-s + (0.0569 − 0.0986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845137 - 0.719436i\)
\(L(\frac12)\) \(\approx\) \(0.845137 - 0.719436i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 1.21i)T \)
7 \( 1 + (0.491 + 2.59i)T \)
13 \( 1 + (3.58 + 0.380i)T \)
good5 \( 1 + (0.473 + 1.76i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.98 + 2.98i)T + 11iT^{2} \)
17 \( 1 + (-0.234 + 0.406i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.932 - 0.932i)T + 19iT^{2} \)
23 \( 1 + (4.27 + 7.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.99 - 1.72i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.05 - 3.95i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-6.47 - 1.73i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.956 + 3.57i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-10.0 + 5.82i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.41 - 2.52i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.77 + 1.60i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.93 - 2.12i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + (0.303 + 0.303i)T + 67iT^{2} \)
71 \( 1 + (7.08 + 1.89i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (5.97 + 1.60i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.79 - 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.80 + 2.80i)T - 83iT^{2} \)
89 \( 1 + (-1.52 + 5.67i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.697 + 2.60i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.54579444599190537963665052006, −10.10754818194682508057744090473, −8.793066593744204936251409433301, −7.83980226729105406952076026754, −6.76475241748763015506332928976, −5.74861546282855692068164841469, −4.83772433520651012578891162882, −4.22693157796830612031835290193, −2.98283374527198061892414126856, −0.53515642195587538746017971589, 2.07606477325135482914264157661, 2.89152269168662483866126533017, 4.59911073060678262229069758341, 5.50137263774767234019480050650, 6.28899495618531103999263005722, 7.34291177367453252142292683751, 7.82861249103418740294111455326, 9.566165327373289212363685762441, 10.28033570534665868000194660969, 11.40995756449669388517220860717

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