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Properties

Label 2-546-273.101-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.953 + 0.300i$
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 + (0.925 + 1.46i)3-s + (−0.499 − 0.866i)4-s + (−2.64 + 1.52i)5-s + (−1.73 + 0.0694i)6-s + (2.61 − 0.391i)7-s + 0.999·8-s + (−1.28 + 2.70i)9-s − 3.05i·10-s − 2.26·11-s + (0.805 − 1.53i)12-s + (−3.11 + 1.81i)13-s + (−0.969 + 2.46i)14-s + (−4.69 − 2.46i)15-s + (−0.5 + 0.866i)16-s + (1.79 + 3.11i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.534 + 0.845i)3-s + (−0.249 − 0.433i)4-s + (−1.18 + 0.684i)5-s + (−0.706 + 0.0283i)6-s + (0.989 − 0.147i)7-s + 0.353·8-s + (−0.429 + 0.903i)9-s − 0.967i·10-s − 0.683·11-s + (0.232 − 0.442i)12-s + (−0.864 + 0.502i)13-s + (−0.259 + 0.657i)14-s + (−1.21 − 0.635i)15-s + (−0.125 + 0.216i)16-s + (0.436 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.953 + 0.300i$
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: \((2,\ 546,\ (\ :1/2),\ -0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118148 - 0.768216i\)
\(L(\frac12)\) \(\approx\) \(0.118148 - 0.768216i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.925 - 1.46i)T \)
7 \( 1 + (-2.61 + 0.391i)T \)
13 \( 1 + (3.11 - 1.81i)T \)
good5 \( 1 + (2.64 - 1.52i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
17 \( 1 + (-1.79 - 3.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 0.827T + 19T^{2} \)
23 \( 1 + (4.00 + 2.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.61 - 4.39i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.78 + 8.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.16 - 0.673i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.03 + 4.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.70 - 8.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.73 - 1.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.44 - 0.834i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.34 + 5.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + 4.06iT - 67T^{2} \)
71 \( 1 + (3.81 - 6.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.74 - 9.94i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.91 - 5.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.79iT - 83T^{2} \)
89 \( 1 + (-7.22 - 4.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.99 - 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07976575110468885283862311997, −10.36473900849391104365323647804, −9.513477658400702043859256809465, −8.279433163297347274825161121384, −7.912681593943113065840331615450, −7.17827659661771677088738238332, −5.67503363952197727881386373948, −4.55795479623664765445605413022, −3.86468286385703219771653084494, −2.36223016753647708368460519649, 0.46255015786276591521061750446, 1.97431053239123700433174556887, 3.20540184375412080647515941875, 4.43941009295602064064191743078, 5.46066169246432935352452727736, 7.28943719662591200493134064422, 7.84907346281880621287837059408, 8.318554160329372880203642802785, 9.261967182493005181405017702164, 10.35078216675205709083708566004

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