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Properties

Label 2-546-7.4-c1-0-13
Degree $2$
Conductor $546$
Sign $-0.991 + 0.126i$
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 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (2.5 − 4.33i)11-s + (0.499 + 0.866i)12-s − 13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−3.5 + 6.06i)17-s + (−0.499 + 0.866i)18-s + (−3.5 − 6.06i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.753 − 1.30i)11-s + (0.144 + 0.249i)12-s − 0.277·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.848 + 1.47i)17-s + (−0.117 + 0.204i)18-s + (−0.802 − 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0485470 - 0.765000i\)
\(L(\frac12)\) \(\approx\) \(0.0485470 - 0.765000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−10.65989570045205121531432061604, −9.203568794378694818929019739795, −8.920030904475143983413660957657, −7.85979638002388090504422929786, −6.67734413253206016608384020228, −6.09817583210124852133315326507, −4.26515255943360664778161075487, −3.36354974527562499121881901580, −2.16862647203008153376810629194, −0.46035840243235644615126640601, 2.10050992828243298710176986334, 3.64135852906212428409458204719, 4.69201831469839573363367913387, 5.79347851888772012721116922892, 6.85859028368445848078516211790, 7.50567041258004675739322081623, 8.814066218781846911788144475148, 9.483528730192644470497644354109, 9.885209671239766446295193912485, 11.03636167687917677423150794342

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