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Properties

Label 2-546-273.38-c1-0-10
Degree $2$
Conductor $546$
Sign $-0.927 - 0.373i$
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 + (1.57 + 0.714i)3-s + (−0.499 + 0.866i)4-s + (−2.19 + 1.26i)5-s + (0.170 + 1.72i)6-s + (−2.61 − 0.373i)7-s − 0.999·8-s + (1.97 + 2.25i)9-s + (−2.19 − 1.26i)10-s + (−1.32 + 2.29i)11-s + (−1.40 + 1.00i)12-s + (3.59 − 0.207i)13-s + (−0.986 − 2.45i)14-s + (−4.37 + 0.432i)15-s + (−0.5 − 0.866i)16-s + (−1.84 + 3.19i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.911 + 0.412i)3-s + (−0.249 + 0.433i)4-s + (−0.982 + 0.567i)5-s + (0.0695 + 0.703i)6-s + (−0.989 − 0.141i)7-s − 0.353·8-s + (0.659 + 0.751i)9-s + (−0.694 − 0.401i)10-s + (−0.399 + 0.691i)11-s + (−0.406 + 0.291i)12-s + (0.998 − 0.0575i)13-s + (−0.263 − 0.656i)14-s + (−1.12 + 0.111i)15-s + (−0.125 − 0.216i)16-s + (−0.447 + 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.927 - 0.373i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280053 + 1.44674i\)
\(L(\frac12)\) \(\approx\) \(0.280053 + 1.44674i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 0.714i)T \)
7 \( 1 + (2.61 + 0.373i)T \)
13 \( 1 + (-3.59 + 0.207i)T \)
good5 \( 1 + (2.19 - 1.26i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.32 - 2.29i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.84 - 3.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.241 + 0.418i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.09 - 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.93iT - 29T^{2} \)
31 \( 1 + (2.71 - 4.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.90 + 3.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.03iT - 41T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 + (-6.66 + 3.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.13 - 4.69i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.98 - 2.87i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.25 + 2.45i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.38 + 0.797i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (6.57 - 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.75 + 6.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.10iT - 83T^{2} \)
89 \( 1 + (-2.37 + 1.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.91T + 97T^{2} \)
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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

−11.05030322338604791770940512397, −10.22304944755449918045671826954, −9.309540286185306436724533746382, −8.340100483932406512808271487676, −7.60989052638491334163993544644, −6.85361867301691136909559247774, −5.74785003380371923261331753314, −4.07802291414003216061166130322, −3.82573559172426222277887021862, −2.56829362206432006336410369671, 0.68345744554646862078244580287, 2.50796382177265592194664233330, 3.54609713721695266099285438039, 4.22359970644116224302263685840, 5.76217501478764221121971408411, 6.81848221701803250285559119759, 7.953131307248154588705693728297, 8.719545221353498600101100911090, 9.363972158083869100215190332483, 10.47017777229284296364018055232

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