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Properties

Label 2-546-273.173-c1-0-36
Degree $2$
Conductor $546$
Sign $-0.728 + 0.685i$
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 + (0.786 − 1.54i)3-s + (−0.499 + 0.866i)4-s + (−1.98 − 1.14i)5-s + (1.72 − 0.0900i)6-s + (−2.60 − 0.487i)7-s − 0.999·8-s + (−1.76 − 2.42i)9-s − 2.28i·10-s + 0.297·11-s + (0.942 + 1.45i)12-s + (−3.20 + 1.66i)13-s + (−0.877 − 2.49i)14-s + (−3.32 + 2.15i)15-s + (−0.5 − 0.866i)16-s + (0.446 − 0.773i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.454 − 0.890i)3-s + (−0.249 + 0.433i)4-s + (−0.886 − 0.511i)5-s + (0.706 − 0.0367i)6-s + (−0.982 − 0.184i)7-s − 0.353·8-s + (−0.587 − 0.809i)9-s − 0.723i·10-s + 0.0895·11-s + (0.272 + 0.419i)12-s + (−0.887 + 0.460i)13-s + (−0.234 − 0.667i)14-s + (−0.858 + 0.557i)15-s + (−0.125 − 0.216i)16-s + (0.108 − 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.728 + 0.685i$
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: \((2,\ 546,\ (\ :1/2),\ -0.728 + 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235742 - 0.594553i\)
\(L(\frac12)\) \(\approx\) \(0.235742 - 0.594553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.786 + 1.54i)T \)
7 \( 1 + (2.60 + 0.487i)T \)
13 \( 1 + (3.20 - 1.66i)T \)
good5 \( 1 + (1.98 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.297T + 11T^{2} \)
17 \( 1 + (-0.446 + 0.773i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.89T + 19T^{2} \)
23 \( 1 + (-6.76 + 3.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.980 + 0.566i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.839 - 1.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.32 + 2.49i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.52 + 3.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.94 + 3.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.21 - 3.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.28 + 3.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.21 + 3.01i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.44iT - 61T^{2} \)
67 \( 1 - 3.39iT - 67T^{2} \)
71 \( 1 + (1.14 + 1.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.16 + 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.46 + 7.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.54iT - 83T^{2} \)
89 \( 1 + (-12.3 + 7.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.19 - 2.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.44654829160626951708156136522, −9.093523984458518469834252649843, −8.644816877948825633082244736570, −7.57719110344617562724490016908, −6.91599009400068855547417605248, −6.17432697078779114649121942626, −4.71048257999526157059085199723, −3.75058736471726588071912011571, −2.51675927058392307206919934006, −0.29271049374661981106178542127, 2.56485616141099072876584552232, 3.36495761664343003642467527294, 4.20118330057186684606190865002, 5.27541481538976678749186177760, 6.53283765140579031289361783281, 7.66172286698426823029182728457, 8.725730913981405391805138152970, 9.564872793337202025331419867640, 10.33973383445162516712128373705, 11.01762510966980198264288290092

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