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Properties

Label 2-546-273.251-c1-0-17
Degree $2$
Conductor $546$
Sign $0.730 + 0.682i$
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 + 1.65i)3-s + (−0.499 − 0.866i)4-s + 0.792i·5-s + (1.18 + 1.26i)6-s + (0.5 − 2.59i)7-s − 0.999·8-s + (−2.5 − 1.65i)9-s + (0.686 + 0.396i)10-s + (2.18 − 3.78i)11-s + (1.68 − 0.396i)12-s + (3.5 − 0.866i)13-s + (−2 − 1.73i)14-s + (−1.31 − 0.396i)15-s + (−0.5 + 0.866i)16-s + (2.18 + 3.78i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.957i)3-s + (−0.249 − 0.433i)4-s + 0.354i·5-s + (0.484 + 0.515i)6-s + (0.188 − 0.981i)7-s − 0.353·8-s + (−0.833 − 0.552i)9-s + (0.216 + 0.125i)10-s + (0.659 − 1.14i)11-s + (0.486 − 0.114i)12-s + (0.970 − 0.240i)13-s + (−0.534 − 0.462i)14-s + (−0.339 − 0.102i)15-s + (−0.125 + 0.216i)16-s + (0.530 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.730 + 0.682i$
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: \((2,\ 546,\ (\ :1/2),\ 0.730 + 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49513 - 0.589542i\)
\(L(\frac12)\) \(\approx\) \(1.49513 - 0.589542i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 1.65i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 0.792iT - 5T^{2} \)
11 \( 1 + (-2.18 + 3.78i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.18 - 3.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.18 + 2.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.68 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.18 - 1.26i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 + (10.1 + 5.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.18 + 4.72i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.939iT - 47T^{2} \)
53 \( 1 - 2.22iT - 53T^{2} \)
59 \( 1 + (5.31 - 3.06i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.1 - 5.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.05 + 13.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 9.62T + 79T^{2} \)
83 \( 1 + 1.58iT - 83T^{2} \)
89 \( 1 + (9.30 + 5.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.372 + 0.644i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.69182203383642355192447857645, −10.29759764663416304252129859135, −9.017208570568371541836826999242, −8.389388240099634917255945399007, −6.78419804036510481825922326936, −5.93268575811222532677011065666, −4.85807504380340642625059993047, −3.73304620807696545768218450700, −3.26030631328793554358354183050, −1.03266414855375778945310815257, 1.48874576662674745439857809010, 2.95152889460607902624652597497, 4.64243794126639984351978408862, 5.40754270409293217259494689245, 6.50645569732145907421714534583, 7.00278946418370196944288660259, 8.316165445064589632614062515176, 8.697779429956535260643388699494, 9.873172556753613301319925247579, 11.27942484763488437905682960345

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