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Properties

Label 2-546-273.17-c1-0-27
Degree $2$
Conductor $546$
Sign $-0.754 + 0.656i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−1.5 + 0.866i)3-s + 4-s + (−3 + 1.73i)5-s + (−1.5 + 0.866i)6-s + (−0.5 + 2.59i)7-s + 8-s + (1.5 − 2.59i)9-s + (−3 + 1.73i)10-s + (−3 − 5.19i)11-s + (−1.5 + 0.866i)12-s + (−2.5 − 2.59i)13-s + (−0.5 + 2.59i)14-s + (3 − 5.19i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + (−1.34 + 0.774i)5-s + (−0.612 + 0.353i)6-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.948 + 0.547i)10-s + (−0.904 − 1.56i)11-s + (−0.433 + 0.249i)12-s + (−0.693 − 0.720i)13-s + (−0.133 + 0.694i)14-s + (0.774 − 1.34i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.754 + 0.656i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.754 + 0.656i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + (6 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.66iT - 37T^{2} \)
41 \( 1 + (9 + 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 - 3.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 1.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 3.46iT - 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.90952866062036696354132516746, −9.881803816249373632320994045217, −8.560900719365241778792062309395, −7.58244845196947454945730049886, −6.65407486869489676414617714753, −5.55371251653496028374813728760, −5.02291177319682035734253706545, −3.48011723194157082190756536934, −3.02099837662712722904490683369, 0, 1.84525002752373343507338707616, 3.84841713519663730346958411568, 4.59844033316176412965575440965, 5.27733040700533496377133635086, 6.77259000409162600257752862291, 7.50705575535959942262475892890, 7.86224625421357269552712601822, 9.634296246168040498540408555709, 10.51650103364840739666380023426

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