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Properties

Label 2-546-273.101-c1-0-16
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: $\chi_{546} (101, \cdot )$
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

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

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