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Properties

Label 2-546-273.173-c1-0-35
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: $\chi_{546} (173, \cdot )$
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

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

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