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  + 0.924·2-s + 2.22·3-s − 1.14·4-s − 3.95·5-s + 2.05·6-s − 3.08·7-s − 2.90·8-s + 1.94·9-s − 3.65·10-s − 3.50·11-s − 2.54·12-s + 3.47·13-s − 2.84·14-s − 8.79·15-s − 0.399·16-s − 3.04·17-s + 1.79·18-s + 7.32·19-s + 4.52·20-s − 6.84·21-s − 3.23·22-s − 6.16·23-s − 6.46·24-s + 10.6·25-s + 3.21·26-s − 2.35·27-s + 3.52·28-s + ⋯
L(s)  = 1  + 0.653·2-s + 1.28·3-s − 0.572·4-s − 1.76·5-s + 0.839·6-s − 1.16·7-s − 1.02·8-s + 0.647·9-s − 1.15·10-s − 1.05·11-s − 0.734·12-s + 0.963·13-s − 0.761·14-s − 2.26·15-s − 0.0999·16-s − 0.739·17-s + 0.423·18-s + 1.68·19-s + 1.01·20-s − 1.49·21-s − 0.690·22-s − 1.28·23-s − 1.31·24-s + 2.12·25-s + 0.629·26-s − 0.452·27-s + 0.666·28-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 \( 1 - 0.924T + 2T^{2} \)
3 \( 1 - 2.22T + 3T^{2} \)
5 \( 1 + 3.95T + 5T^{2} \)
7 \( 1 + 3.08T + 7T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 - 3.47T + 13T^{2} \)
17 \( 1 + 3.04T + 17T^{2} \)
19 \( 1 - 7.32T + 19T^{2} \)
23 \( 1 + 6.16T + 23T^{2} \)
29 \( 1 + 3.33T + 29T^{2} \)
31 \( 1 - 5.01T + 31T^{2} \)
37 \( 1 + 9.39T + 37T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 - 0.457T + 43T^{2} \)
47 \( 1 + 4.87T + 47T^{2} \)
53 \( 1 - 0.519T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 2.71T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 6.36T + 71T^{2} \)
73 \( 1 + 1.19T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 7.25T + 97T^{2} \)
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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

−10.20418821280149444641977313749, −9.271532439431975455356960626771, −8.424202700002752156431602759340, −7.949030497087308098263455845304, −6.88934649274530555031312731447, −5.49995678723189486526524048832, −4.18811218614157212377796585167, −3.48332742690451705926269879720, −2.97440946318885662155519275568, 0, 2.97440946318885662155519275568, 3.48332742690451705926269879720, 4.18811218614157212377796585167, 5.49995678723189486526524048832, 6.88934649274530555031312731447, 7.949030497087308098263455845304, 8.424202700002752156431602759340, 9.271532439431975455356960626771, 10.20418821280149444641977313749

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