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Properties

Label 2-546-273.173-c1-0-20
Degree $2$
Conductor $546$
Sign $0.370 + 0.928i$
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 + (0.458 − 1.67i)3-s + (−0.499 + 0.866i)4-s + (1.57 + 0.908i)5-s + (−1.67 + 0.438i)6-s + (−0.414 + 2.61i)7-s + 0.999·8-s + (−2.58 − 1.53i)9-s − 1.81i·10-s + 2.05·11-s + (1.21 + 1.23i)12-s + (3.57 + 0.463i)13-s + (2.47 − 0.947i)14-s + (2.23 − 2.21i)15-s + (−0.5 − 0.866i)16-s + (2.84 − 4.92i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.264 − 0.964i)3-s + (−0.249 + 0.433i)4-s + (0.703 + 0.406i)5-s + (−0.684 + 0.178i)6-s + (−0.156 + 0.987i)7-s + 0.353·8-s + (−0.860 − 0.510i)9-s − 0.574i·10-s + 0.618·11-s + (0.351 + 0.355i)12-s + (0.991 + 0.128i)13-s + (0.660 − 0.253i)14-s + (0.577 − 0.571i)15-s + (−0.125 − 0.216i)16-s + (0.689 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.370 + 0.928i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.370 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24641 - 0.844814i\)
\(L(\frac12)\) \(\approx\) \(1.24641 - 0.844814i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.458 + 1.67i)T \)
7 \( 1 + (0.414 - 2.61i)T \)
13 \( 1 + (-3.57 - 0.463i)T \)
good5 \( 1 + (-1.57 - 0.908i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.05T + 11T^{2} \)
17 \( 1 + (-2.84 + 4.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.59T + 19T^{2} \)
23 \( 1 + (-7.56 + 4.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.724 - 0.418i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.97 - 6.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.65 - 2.10i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.397 + 0.229i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.836 - 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.94 + 1.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.497 - 0.286i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.89 + 1.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.89iT - 61T^{2} \)
67 \( 1 + 5.15iT - 67T^{2} \)
71 \( 1 + (-4.24 - 7.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.11 + 8.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.68 - 6.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.0iT - 83T^{2} \)
89 \( 1 + (3.34 - 1.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.72 - 2.98i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.72627061555752731731400482294, −9.589690526540137185439154234367, −8.891604510938157008646052099551, −8.255195767195053001918952892907, −6.87298449764136180355430057981, −6.32756393447941649380739366232, −5.12494490315444456774285774113, −3.26182879110052833753349729011, −2.49955437293704555102903479544, −1.24440792761629504689347913152, 1.36524720393111322665900127476, 3.45957003968599609967117904091, 4.33510185131169195772548276847, 5.52483934824377720559905448474, 6.26057454481357172790223530480, 7.50245437230410716934038150178, 8.505328738614318692651252531077, 9.196397297165175327029944249885, 9.972216272723413586822370992696, 10.63609250976658219384784501564

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