Properties

Degree $2$
Conductor $547$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.826·2-s + 2.26·3-s − 1.31·4-s − 0.786·5-s − 1.86·6-s − 5.06·7-s + 2.74·8-s + 2.10·9-s + 0.649·10-s + 3.16·11-s − 2.97·12-s − 2.09·13-s + 4.18·14-s − 1.77·15-s + 0.370·16-s − 0.108·17-s − 1.74·18-s − 6.43·19-s + 1.03·20-s − 11.4·21-s − 2.61·22-s − 2.45·23-s + 6.19·24-s − 4.38·25-s + 1.72·26-s − 2.01·27-s + 6.66·28-s + ⋯
L(s)  = 1  − 0.584·2-s + 1.30·3-s − 0.658·4-s − 0.351·5-s − 0.762·6-s − 1.91·7-s + 0.968·8-s + 0.703·9-s + 0.205·10-s + 0.954·11-s − 0.859·12-s − 0.580·13-s + 1.11·14-s − 0.458·15-s + 0.0927·16-s − 0.0264·17-s − 0.410·18-s − 1.47·19-s + 0.231·20-s − 2.49·21-s − 0.557·22-s − 0.511·23-s + 1.26·24-s − 0.876·25-s + 0.338·26-s − 0.387·27-s + 1.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(547\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{547} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 547,\ (\ :1/2),\ -1)\)

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)$
bad547 \( 1 + T \)
good2 \( 1 + 0.826T + 2T^{2} \)
3 \( 1 - 2.26T + 3T^{2} \)
5 \( 1 + 0.786T + 5T^{2} \)
7 \( 1 + 5.06T + 7T^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 + 2.09T + 13T^{2} \)
17 \( 1 + 0.108T + 17T^{2} \)
19 \( 1 + 6.43T + 19T^{2} \)
23 \( 1 + 2.45T + 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + 4.03T + 31T^{2} \)
37 \( 1 + 5.74T + 37T^{2} \)
41 \( 1 - 9.42T + 41T^{2} \)
43 \( 1 - 6.98T + 43T^{2} \)
47 \( 1 - 4.86T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 - 5.35T + 59T^{2} \)
61 \( 1 - 6.38T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 0.423T + 79T^{2} \)
83 \( 1 - 2.31T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 0.251T + 97T^{2} \)
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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

−9.810915045747187976754306583320, −9.379288162460641723714581863049, −8.860295872510546986892083044453, −7.87475164349536068927544988809, −7.04275489755061705632699743264, −5.91996445594109864800340048987, −4.04244162173816763074343180083, −3.65413467526050464510364583609, −2.23663710441619866483785466508, 0, 2.23663710441619866483785466508, 3.65413467526050464510364583609, 4.04244162173816763074343180083, 5.91996445594109864800340048987, 7.04275489755061705632699743264, 7.87475164349536068927544988809, 8.860295872510546986892083044453, 9.379288162460641723714581863049, 9.810915045747187976754306583320

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