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Properties

Label 2-546-273.257-c1-0-26
Degree $2$
Conductor $546$
Sign $0.404 + 0.914i$
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  − 2-s + (−0.565 + 1.63i)3-s + 4-s + (1.26 + 0.730i)5-s + (0.565 − 1.63i)6-s + (−1.08 − 2.41i)7-s − 8-s + (−2.36 − 1.85i)9-s + (−1.26 − 0.730i)10-s + (1.75 − 3.04i)11-s + (−0.565 + 1.63i)12-s + (0.214 − 3.59i)13-s + (1.08 + 2.41i)14-s + (−1.90 + 1.65i)15-s + 16-s − 7.58·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.326 + 0.945i)3-s + 0.5·4-s + (0.565 + 0.326i)5-s + (0.230 − 0.668i)6-s + (−0.411 − 0.911i)7-s − 0.353·8-s + (−0.787 − 0.616i)9-s + (−0.399 − 0.230i)10-s + (0.530 − 0.918i)11-s + (−0.163 + 0.472i)12-s + (0.0595 − 0.998i)13-s + (0.290 + 0.644i)14-s + (−0.493 + 0.428i)15-s + 0.250·16-s − 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.404 + 0.914i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.404 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562928 - 0.366737i\)
\(L(\frac12)\) \(\approx\) \(0.562928 - 0.366737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.565 - 1.63i)T \)
7 \( 1 + (1.08 + 2.41i)T \)
13 \( 1 + (-0.214 + 3.59i)T \)
good5 \( 1 + (-1.26 - 0.730i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.75 + 3.04i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 + (1.72 + 2.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.60iT - 23T^{2} \)
29 \( 1 + (0.170 - 0.0985i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.34 - 9.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.56iT - 37T^{2} \)
41 \( 1 + (-2.60 + 1.50i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.61 + 9.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (11.0 + 6.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.21 + 2.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.30iT - 59T^{2} \)
61 \( 1 + (-0.865 + 0.499i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.78 - 2.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.33 + 9.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.94 + 5.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.174 + 0.302i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.72iT - 83T^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 + (1.72 - 2.99i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64760261653999159485085602122, −9.931213028025633140286929443465, −8.938923651098441716077133664464, −8.338465563426486369377517836786, −6.65432902427639154138663129422, −6.41648752317977117532989892655, −5.00890077504630644186165391276, −3.80448178046098470455122998249, −2.68116811054520422112548698549, −0.49013719636227857728370232618, 1.70045548346239026367097673860, 2.38782715519072186562883735012, 4.43793175901731807337735465653, 5.89702755681152519186449904000, 6.40780040387749808355962317805, 7.31877727789886192539312122305, 8.403080320532748201126720341575, 9.273445387053993164671808942067, 9.716668499427206727575819948584, 11.28149887638792216506915962863

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