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Properties

Label 2-546-273.272-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.455 - 0.890i$
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 + (−0.420 − 1.68i)3-s + 4-s + 3.36i·5-s + (0.420 + 1.68i)6-s + (2.37 + 1.16i)7-s − 8-s + (−2.64 + 1.41i)9-s − 3.36i·10-s + (−0.420 − 1.68i)12-s + (−2.79 + 2.27i)13-s + (−2.37 − 1.16i)14-s + (5.64 − 1.41i)15-s + 16-s − 7.82·17-s + (2.64 − 1.41i)18-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.242 − 0.970i)3-s + 0.5·4-s + 1.50i·5-s + (0.171 + 0.685i)6-s + (0.898 + 0.439i)7-s − 0.353·8-s + (−0.881 + 0.471i)9-s − 1.06i·10-s + (−0.121 − 0.485i)12-s + (−0.775 + 0.631i)13-s + (−0.635 − 0.311i)14-s + (1.45 − 0.365i)15-s + 0.250·16-s − 1.89·17-s + (0.623 − 0.333i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.455 - 0.890i$
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: \((2,\ 546,\ (\ :1/2),\ -0.455 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.271635 + 0.444114i\)
\(L(\frac12)\) \(\approx\) \(0.271635 + 0.444114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.420 + 1.68i)T \)
7 \( 1 + (-2.37 - 1.16i)T \)
13 \( 1 + (2.79 - 2.27i)T \)
good5 \( 1 - 3.36iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 7.82T + 17T^{2} \)
19 \( 1 + 5.59T + 19T^{2} \)
23 \( 1 + 0.500iT - 23T^{2} \)
29 \( 1 + 5.15iT - 29T^{2} \)
31 \( 1 + 3.06T + 31T^{2} \)
37 \( 1 - 2.32iT - 37T^{2} \)
41 \( 1 - 9.87iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 4.33iT - 47T^{2} \)
53 \( 1 + 0.500iT - 53T^{2} \)
59 \( 1 + 2.16iT - 59T^{2} \)
61 \( 1 + 4.55iT - 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + 6.58T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 0.708T + 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + 3.14iT - 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−11.25069925732348090600473178106, −10.45930765397435892320130889076, −9.193115361478112306427773639814, −8.286500513948373891461527869982, −7.44594760272247832429128521625, −6.67597151198996466831239697267, −6.10980913155581863743827099506, −4.51935334607244410788646319847, −2.55179049973386911530200234851, −2.06149969785040731582345893104, 0.36000312018704928925885794995, 2.10854054547831715185297251060, 4.08828198338155360025778847272, 4.79110417050554257212035033348, 5.62916517889122041407504529458, 7.05424924161608351015621336489, 8.289397551846292355164804077957, 8.799976869680243976535923856317, 9.441669810331042849279698213558, 10.68812981300513494172384576286

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