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Properties

Label 2-546-273.152-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.146 - 0.989i$
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  + (−0.866 + 0.5i)2-s + (−0.685 − 1.59i)3-s + (0.499 − 0.866i)4-s + (−0.587 + 1.01i)5-s + (1.38 + 1.03i)6-s + (−1.73 − 1.99i)7-s + 0.999i·8-s + (−2.05 + 2.18i)9-s − 1.17i·10-s − 0.918i·11-s + (−1.72 − 0.201i)12-s + (−0.207 + 3.59i)13-s + (2.50 + 0.859i)14-s + (2.02 + 0.236i)15-s + (−0.5 − 0.866i)16-s + (−1.08 + 1.88i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.395 − 0.918i)3-s + (0.249 − 0.433i)4-s + (−0.262 + 0.454i)5-s + (0.567 + 0.422i)6-s + (−0.656 − 0.754i)7-s + 0.353i·8-s + (−0.686 + 0.727i)9-s − 0.371i·10-s − 0.277i·11-s + (−0.496 − 0.0581i)12-s + (−0.0575 + 0.998i)13-s + (0.668 + 0.229i)14-s + (0.521 + 0.0611i)15-s + (−0.125 − 0.216i)16-s + (−0.263 + 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.146 - 0.989i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262883 + 0.304655i\)
\(L(\frac12)\) \(\approx\) \(0.262883 + 0.304655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.685 + 1.59i)T \)
7 \( 1 + (1.73 + 1.99i)T \)
13 \( 1 + (0.207 - 3.59i)T \)
good5 \( 1 + (0.587 - 1.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.918iT - 11T^{2} \)
17 \( 1 + (1.08 - 1.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 0.0298iT - 19T^{2} \)
23 \( 1 + (2.48 - 1.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0565 + 0.0326i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.64 + 2.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.47 - 2.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.01 - 6.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.74 - 4.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.53 - 11.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.75 + 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.80 - 3.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 0.759iT - 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 + (-8.16 + 4.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.34 - 1.35i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.71 + 6.42i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (5.01 + 8.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.74 - 5.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.10968086005091422333278130413, −10.18324665907838751366959562423, −9.244728702280075765379616290078, −8.124398436679847491543172950341, −7.39517819127379695246131843045, −6.58868189986525237256334653511, −6.06187262803238299386098960991, −4.50667097111787848050806520573, −2.99168006277111653347760172873, −1.41561085690163519488210451030, 0.30761348018734560268346047491, 2.58342409788511248591867757403, 3.67951053231812185069540795967, 4.88974945928953282167561010004, 5.83421767618361154716411058156, 6.93670040675041262896656632118, 8.344214702846461792369139610203, 8.852324203874141559110249674696, 9.854783204698693913867617081545, 10.31575302036555266672029021898

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