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Properties

Label 2-546-273.62-c1-0-13
Degree $2$
Conductor $546$
Sign $-0.505 - 0.862i$
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 + (1.48 + 0.889i)3-s + (−0.499 + 0.866i)4-s − 0.655i·5-s + (−0.0267 + 1.73i)6-s + (−2.06 + 1.65i)7-s − 0.999·8-s + (1.41 + 2.64i)9-s + (0.567 − 0.327i)10-s + (0.366 + 0.635i)11-s + (−1.51 + 0.842i)12-s + (−0.424 + 3.58i)13-s + (−2.46 − 0.958i)14-s + (0.582 − 0.974i)15-s + (−0.5 − 0.866i)16-s + (0.581 − 1.00i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.858 + 0.513i)3-s + (−0.249 + 0.433i)4-s − 0.293i·5-s + (−0.0109 + 0.707i)6-s + (−0.779 + 0.626i)7-s − 0.353·8-s + (0.473 + 0.881i)9-s + (0.179 − 0.103i)10-s + (0.110 + 0.191i)11-s + (−0.436 + 0.243i)12-s + (−0.117 + 0.993i)13-s + (−0.659 − 0.256i)14-s + (0.150 − 0.251i)15-s + (−0.125 − 0.216i)16-s + (0.141 − 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.505 - 0.862i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991185 + 1.73001i\)
\(L(\frac12)\) \(\approx\) \(0.991185 + 1.73001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 0.889i)T \)
7 \( 1 + (2.06 - 1.65i)T \)
13 \( 1 + (0.424 - 3.58i)T \)
good5 \( 1 + 0.655iT - 5T^{2} \)
11 \( 1 + (-0.366 - 0.635i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.581 + 1.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.51 - 4.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.55 + 3.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.839 - 0.484i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.20T + 31T^{2} \)
37 \( 1 + (-3.59 + 2.07i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.22 + 2.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.81 + 6.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.48iT - 47T^{2} \)
53 \( 1 + 1.29iT - 53T^{2} \)
59 \( 1 + (2.15 + 1.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.932 + 0.538i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.07 - 2.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.09 + 7.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.93T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 + (-12.6 + 7.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.36 - 5.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.99868414091479308803741286669, −9.929629194858363546000842397861, −9.021685090816844492484190751629, −8.720930402251296652071968048544, −7.46405615141860390314705286393, −6.62166129642584164269017582967, −5.45463657266665128434377870626, −4.44694971515380700834687784101, −3.49003070551392635891473916856, −2.27660981579494483648950909500, 0.991256139675666856186240536827, 2.74167685453296282026516423315, 3.31747993302589968625246600117, 4.51606170374736521241084440319, 5.99784578124568691933258315357, 6.93858264645901455515902733314, 7.75487860626646186606764013057, 8.951433166538443905986804986136, 9.597834773033538820587642796882, 10.59399189808106017325895553215

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