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  + 1.52·2-s − 2.58·3-s + 0.316·4-s + 1.24·5-s − 3.93·6-s − 0.899·7-s − 2.56·8-s + 3.68·9-s + 1.89·10-s + 3.77·11-s − 0.819·12-s − 4.29·13-s − 1.36·14-s − 3.22·15-s − 4.53·16-s − 7.86·17-s + 5.61·18-s − 6.61·19-s + 0.395·20-s + 2.32·21-s + 5.75·22-s + 3.37·23-s + 6.62·24-s − 3.44·25-s − 6.53·26-s − 1.78·27-s − 0.285·28-s + ⋯
L(s)  = 1  + 1.07·2-s − 1.49·3-s + 0.158·4-s + 0.557·5-s − 1.60·6-s − 0.339·7-s − 0.905·8-s + 1.22·9-s + 0.600·10-s + 1.13·11-s − 0.236·12-s − 1.18·13-s − 0.365·14-s − 0.832·15-s − 1.13·16-s − 1.90·17-s + 1.32·18-s − 1.51·19-s + 0.0883·20-s + 0.507·21-s + 1.22·22-s + 0.704·23-s + 1.35·24-s − 0.688·25-s − 1.28·26-s − 0.343·27-s − 0.0538·28-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 \( 1 - 1.52T + 2T^{2} \)
3 \( 1 + 2.58T + 3T^{2} \)
5 \( 1 - 1.24T + 5T^{2} \)
7 \( 1 + 0.899T + 7T^{2} \)
11 \( 1 - 3.77T + 11T^{2} \)
13 \( 1 + 4.29T + 13T^{2} \)
17 \( 1 + 7.86T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 + 2.81T + 29T^{2} \)
31 \( 1 + 1.72T + 31T^{2} \)
37 \( 1 + 4.86T + 37T^{2} \)
41 \( 1 - 5.86T + 41T^{2} \)
43 \( 1 - 7.30T + 43T^{2} \)
47 \( 1 - 2.08T + 47T^{2} \)
53 \( 1 - 3.26T + 53T^{2} \)
59 \( 1 + 6.21T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 - 5.15T + 79T^{2} \)
83 \( 1 - 6.55T + 83T^{2} \)
89 \( 1 + 4.60T + 89T^{2} \)
97 \( 1 - 9.16T + 97T^{2} \)
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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

−10.71825302217784913200482186745, −9.536231123722122125089894829919, −8.890360020130550599050498701657, −6.93605190441621134166151448546, −6.41486283066228522962352600611, −5.68242795770982679480897279594, −4.70879646269567418472658766508, −4.06998371142865556298482755455, −2.27475206779672789357193027850, 0, 2.27475206779672789357193027850, 4.06998371142865556298482755455, 4.70879646269567418472658766508, 5.68242795770982679480897279594, 6.41486283066228522962352600611, 6.93605190441621134166151448546, 8.890360020130550599050498701657, 9.536231123722122125089894829919, 10.71825302217784913200482186745

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