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Properties

Label 2-546-13.3-c1-0-9
Degree $2$
Conductor $546$
Sign $0.0128 - 0.999i$
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  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 4·5-s + (−0.499 + 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (2 + 3.46i)10-s + (1.5 + 2.59i)11-s − 0.999·12-s + (−2.5 − 2.59i)13-s − 0.999·14-s + (2 + 3.46i)15-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.78·5-s + (−0.204 + 0.353i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.632 + 1.09i)10-s + (0.452 + 0.783i)11-s − 0.288·12-s + (−0.693 − 0.720i)13-s − 0.267·14-s + (0.516 + 0.894i)15-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67330 + 1.65198i\)
\(L(\frac12)\) \(\approx\) \(1.67330 + 1.65198i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good5 \( 1 - 4T + 5T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 11T + 53T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 + 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.65234761240300713044289414016, −9.909368445830808424276765756360, −9.387213486985284318908274632670, −8.503923422612367660369418439499, −7.23088041632629343234814940258, −6.32663603249249366271002008719, −5.40245095979644550487900414957, −4.77580161198113373617957070828, −3.15942315338852056502104693645, −2.06651423055000875473771703220, 1.41657840200847134643381722163, 2.32812910948998003646911401521, 3.56419567736335344125202481975, 4.98948553865467236865492509446, 6.12047470895864318007447049586, 6.51905856959987526073644511782, 8.013122467384839701530357397457, 9.182987224855806064294159330136, 9.702530217671609123200659693485, 10.48036719555506605811388077778

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