Properties

Label 2-546-273.272-c1-0-6
Degree $2$
Conductor $546$
Sign $-0.744 - 0.667i$
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 + (1.26 + 1.18i)3-s + 4-s + 3.37i·5-s + (−1.26 − 1.18i)6-s + (−2.52 + 0.792i)7-s − 8-s + (0.186 + 2.99i)9-s − 3.37i·10-s + 5.74·11-s + (1.26 + 1.18i)12-s + (−3.46 − i)13-s + (2.52 − 0.792i)14-s + (−4 + 4.25i)15-s + 16-s − 0.792·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.728 + 0.684i)3-s + 0.5·4-s + 1.50i·5-s + (−0.515 − 0.484i)6-s + (−0.954 + 0.299i)7-s − 0.353·8-s + (0.0620 + 0.998i)9-s − 1.06i·10-s + 1.73·11-s + (0.364 + 0.342i)12-s + (−0.960 − 0.277i)13-s + (0.674 − 0.211i)14-s + (−1.03 + 1.09i)15-s + 0.250·16-s − 0.192·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395712 + 1.03360i\)
\(L(\frac12)\) \(\approx\) \(0.395712 + 1.03360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 - 1.18i)T \)
7 \( 1 + (2.52 - 0.792i)T \)
13 \( 1 + (3.46 + i)T \)
good5 \( 1 - 3.37iT - 5T^{2} \)
11 \( 1 - 5.74T + 11T^{2} \)
17 \( 1 + 0.792T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
23 \( 1 - 0.147iT - 23T^{2} \)
29 \( 1 - 2.37iT - 29T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 + 8.36iT - 37T^{2} \)
41 \( 1 + 8.37iT - 41T^{2} \)
43 \( 1 + 2.62T + 43T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 - 2.74iT - 59T^{2} \)
61 \( 1 - 7.74iT - 61T^{2} \)
67 \( 1 - 5.98iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 - 6.74iT - 83T^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 - 9.15T + 97T^{2} \)
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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

−10.76343427308720956043079561269, −10.16779269889132917723446173033, −9.384506275690607609318611202662, −8.829152211364820557658593074166, −7.47274157640343991314862257529, −6.84372926648081368371572602283, −5.94873171188308066669010647178, −4.11057208566212940316041648361, −3.15386633984070953913949512485, −2.30322536517924624411873183424, 0.73625931991600220644306005172, 1.92501955632641199576481600557, 3.51214933867541675952484142978, 4.64117772006033055295650584685, 6.35579202780354670805301820231, 6.82446572956648306850781041250, 8.071834528803574785197665435595, 8.720603434903695616245822053100, 9.501921919520597905010171814152, 9.854566136886059509048006278202

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