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  + 0.735·2-s + 0.544·3-s − 1.45·4-s + 0.962·5-s + 0.400·6-s − 3.25·7-s − 2.54·8-s − 2.70·9-s + 0.707·10-s − 0.883·11-s − 0.794·12-s − 4.48·13-s − 2.39·14-s + 0.523·15-s + 1.04·16-s − 3.18·17-s − 1.98·18-s + 4.12·19-s − 1.40·20-s − 1.77·21-s − 0.649·22-s + 3.73·23-s − 1.38·24-s − 4.07·25-s − 3.29·26-s − 3.10·27-s + 4.74·28-s + ⋯
L(s)  = 1  + 0.519·2-s + 0.314·3-s − 0.729·4-s + 0.430·5-s + 0.163·6-s − 1.22·7-s − 0.899·8-s − 0.901·9-s + 0.223·10-s − 0.266·11-s − 0.229·12-s − 1.24·13-s − 0.639·14-s + 0.135·15-s + 0.262·16-s − 0.771·17-s − 0.468·18-s + 0.946·19-s − 0.314·20-s − 0.386·21-s − 0.138·22-s + 0.779·23-s − 0.282·24-s − 0.814·25-s − 0.646·26-s − 0.597·27-s + 0.897·28-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 \( 1 - 0.735T + 2T^{2} \)
3 \( 1 - 0.544T + 3T^{2} \)
5 \( 1 - 0.962T + 5T^{2} \)
7 \( 1 + 3.25T + 7T^{2} \)
11 \( 1 + 0.883T + 11T^{2} \)
13 \( 1 + 4.48T + 13T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 - 4.12T + 19T^{2} \)
23 \( 1 - 3.73T + 23T^{2} \)
29 \( 1 - 6.33T + 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 9.31T + 43T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 + 7.06T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 3.04T + 61T^{2} \)
67 \( 1 + 9.15T + 67T^{2} \)
71 \( 1 + 4.57T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 - 7.20T + 79T^{2} \)
83 \( 1 - 7.39T + 83T^{2} \)
89 \( 1 - 0.838T + 89T^{2} \)
97 \( 1 - 0.475T + 97T^{2} \)
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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

−10.00828728792462251160152786014, −9.509859634862580768198524283937, −8.828391170310390024604574655929, −7.68106061500103327823277636609, −6.48784395151949075309761480761, −5.61363761233904359929176108641, −4.73436831617832172151711616752, −3.37748843144656817619587842063, −2.63646459844130186224277379099, 0, 2.63646459844130186224277379099, 3.37748843144656817619587842063, 4.73436831617832172151711616752, 5.61363761233904359929176108641, 6.48784395151949075309761480761, 7.68106061500103327823277636609, 8.828391170310390024604574655929, 9.509859634862580768198524283937, 10.00828728792462251160152786014

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