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Properties

Label 2-546-273.101-c1-0-23
Degree $2$
Conductor $546$
Sign $-0.979 + 0.203i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.5 + 0.866i)5-s + (1.5 − 0.866i)6-s + (0.5 − 2.59i)7-s + 0.999·8-s + (1.5 + 2.59i)9-s − 1.73i·10-s + 3·11-s + 1.73i·12-s + (1 + 3.46i)13-s + (2 + 1.73i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.670 + 0.387i)5-s + (0.612 − 0.353i)6-s + (0.188 − 0.981i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s − 0.547i·10-s + 0.904·11-s + 0.499i·12-s + (0.277 + 0.960i)13-s + (0.534 + 0.462i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.979 + 0.203i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.979 + 0.203i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (-1 - 3.46i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (3 + 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.5 - 4.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 0.866i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 0.866i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9 - 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.73iT - 61T^{2} \)
67 \( 1 - 1.73iT - 67T^{2} \)
71 \( 1 + (-4.5 + 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.5 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.5 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.62108043940049074657150584349, −9.450187195264585990824451347706, −8.444928812939337793360101341032, −7.29038347918450508018725824538, −6.94732879534821655955302750296, −6.14122045506470045306757991297, −4.69139730742457252405064256904, −3.97965966764672119768840062102, −1.69273878723798108628976202305, 0, 1.88447900803097074366696213508, 3.73424133301301353413046163971, 4.36063974272498269005425692716, 5.68012931751578587431709225936, 6.44388290227135823512277668245, 8.008972840050733078740072962436, 8.677292287839795598772334090122, 9.503195583626647707080266526248, 10.56033277537924069800158751932

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