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Properties

Label 2-546-273.272-c1-0-29
Degree $2$
Conductor $546$
Sign $0.0907 + 0.995i$
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 + (−1.5 − 0.866i)3-s + 4-s − 1.73i·5-s + (−1.5 − 0.866i)6-s + (2.5 − 0.866i)7-s + 8-s + (1.5 + 2.59i)9-s − 1.73i·10-s + (−1.5 − 0.866i)12-s + (−1 − 3.46i)13-s + (2.5 − 0.866i)14-s + (−1.49 + 2.59i)15-s + 16-s − 3·17-s + (1.5 + 2.59i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 − 0.499i)3-s + 0.5·4-s − 0.774i·5-s + (−0.612 − 0.353i)6-s + (0.944 − 0.327i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s − 0.547i·10-s + (−0.433 − 0.249i)12-s + (−0.277 − 0.960i)13-s + (0.668 − 0.231i)14-s + (−0.387 + 0.670i)15-s + 0.250·16-s − 0.727·17-s + (0.353 + 0.612i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0907 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0907 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0907 + 0.995i$
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: \((2,\ 546,\ (\ :1/2),\ 0.0907 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29272 - 1.18024i\)
\(L(\frac12)\) \(\approx\) \(1.29272 - 1.18024i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
13 \( 1 + (1 + 3.46i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 1.73iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 - 12.1iT - 47T^{2} \)
53 \( 1 + 3.46iT - 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 3.46iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.81819987192700384126653738571, −10.08642791535168582302754099414, −8.488621579908768185943012188625, −7.85036162334922702114329437952, −6.77748434292119410393062732674, −5.83743061463752645412166003067, −4.85767348395588746651493645720, −4.37952033965708351220728726435, −2.39770062202245478712646422186, −0.956148636789177514059356906960, 1.91088873287514032072613703014, 3.42847535983241504989019650171, 4.62146132573888650910578520334, 5.19261235971330291783398588582, 6.50379392837217768899571940823, 6.89889442190176206024077455638, 8.300422109758970489867340675834, 9.446308119477983334854180644566, 10.51081536822856523012362329335, 11.12527416843747569924736155900

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