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Properties

Label 2-546-273.173-c1-0-2
Degree $2$
Conductor $546$
Sign $0.531 - 0.847i$
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 + (−1.73 + 0.0276i)3-s + (−0.499 + 0.866i)4-s + (−2.84 − 1.64i)5-s + (0.889 + 1.48i)6-s + (−0.683 − 2.55i)7-s + 0.999·8-s + (2.99 − 0.0958i)9-s + 3.28i·10-s − 2.07·11-s + (0.841 − 1.51i)12-s + (−3.53 − 0.716i)13-s + (−1.87 + 1.86i)14-s + (4.97 + 2.76i)15-s + (−0.5 − 0.866i)16-s + (−0.561 + 0.973i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.999 + 0.0159i)3-s + (−0.249 + 0.433i)4-s + (−1.27 − 0.735i)5-s + (0.363 + 0.606i)6-s + (−0.258 − 0.966i)7-s + 0.353·8-s + (0.999 − 0.0319i)9-s + 1.03i·10-s − 0.624·11-s + (0.243 − 0.436i)12-s + (−0.980 − 0.198i)13-s + (−0.500 + 0.499i)14-s + (1.28 + 0.714i)15-s + (−0.125 − 0.216i)16-s + (−0.136 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.531 - 0.847i$
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: \((2,\ 546,\ (\ :1/2),\ 0.531 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121489 + 0.0671901i\)
\(L(\frac12)\) \(\approx\) \(0.121489 + 0.0671901i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0276i)T \)
7 \( 1 + (0.683 + 2.55i)T \)
13 \( 1 + (3.53 + 0.716i)T \)
good5 \( 1 + (2.84 + 1.64i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
17 \( 1 + (0.561 - 0.973i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.01T + 19T^{2} \)
23 \( 1 + (-3.03 + 1.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.97 - 4.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.86 - 3.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.03 - 3.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.52 - 4.92i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.35 - 5.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.05 + 0.609i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.05 - 2.92i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.49 + 4.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 0.241iT - 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + (3.94 + 6.83i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.878 + 1.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.48 - 4.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.999iT - 83T^{2} \)
89 \( 1 + (6.59 - 3.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.21 - 5.57i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.83348597539482253344217948141, −10.41527171034018806918255055711, −9.370566171861357862854181021915, −8.155333092242515559827169259979, −7.51064133963114315783095651524, −6.57133015996113654545714558916, −4.81977897296634431003947219335, −4.53823416624179580854550242711, −3.19301916983618118409736565114, −1.03062801269681215108038026036, 0.12899311287221378759341188527, 2.62898947674518933013572426208, 4.22713927948730742318002014325, 5.20718063348285038703778320822, 6.18134874806701622713640063054, 7.13581391359997292772546086506, 7.66619083669438990468125698969, 8.794392306099681555355492993639, 9.886998267956431752507666924146, 10.67613894279831323003560300816

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