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  − 1.04·2-s − 2.71·3-s − 0.908·4-s + 0.714·5-s + 2.83·6-s − 2.03·7-s + 3.03·8-s + 4.38·9-s − 0.745·10-s + 5.43·11-s + 2.46·12-s − 3.96·13-s + 2.12·14-s − 1.94·15-s − 1.35·16-s − 1.30·17-s − 4.58·18-s + 8.24·19-s − 0.648·20-s + 5.52·21-s − 5.67·22-s − 2.44·23-s − 8.25·24-s − 4.49·25-s + 4.14·26-s − 3.76·27-s + 1.84·28-s + ⋯
L(s)  = 1  − 0.738·2-s − 1.56·3-s − 0.454·4-s + 0.319·5-s + 1.15·6-s − 0.768·7-s + 1.07·8-s + 1.46·9-s − 0.235·10-s + 1.63·11-s + 0.712·12-s − 1.10·13-s + 0.567·14-s − 0.501·15-s − 0.339·16-s − 0.317·17-s − 1.08·18-s + 1.89·19-s − 0.145·20-s + 1.20·21-s − 1.21·22-s − 0.510·23-s − 1.68·24-s − 0.898·25-s + 0.812·26-s − 0.725·27-s + 0.349·28-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 \( 1 + 1.04T + 2T^{2} \)
3 \( 1 + 2.71T + 3T^{2} \)
5 \( 1 - 0.714T + 5T^{2} \)
7 \( 1 + 2.03T + 7T^{2} \)
11 \( 1 - 5.43T + 11T^{2} \)
13 \( 1 + 3.96T + 13T^{2} \)
17 \( 1 + 1.30T + 17T^{2} \)
19 \( 1 - 8.24T + 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 - 1.49T + 29T^{2} \)
31 \( 1 + 6.02T + 31T^{2} \)
37 \( 1 + 6.36T + 37T^{2} \)
41 \( 1 + 2.87T + 41T^{2} \)
43 \( 1 + 5.76T + 43T^{2} \)
47 \( 1 - 6.42T + 47T^{2} \)
53 \( 1 + 0.932T + 53T^{2} \)
59 \( 1 - 1.57T + 59T^{2} \)
61 \( 1 + 7.93T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 - 9.81T + 73T^{2} \)
79 \( 1 + 2.80T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 1.89T + 89T^{2} \)
97 \( 1 - 0.276T + 97T^{2} \)
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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

−10.01737908528731549939561120849, −9.768259699816330321164151765044, −8.933617499905635007068164653641, −7.42814923511835420874764519567, −6.75557946173021316806256849168, −5.74037497357810135460719075010, −4.90646652801495110987904506458, −3.74867835204588607198789804190, −1.42753079084273670382675661256, 0, 1.42753079084273670382675661256, 3.74867835204588607198789804190, 4.90646652801495110987904506458, 5.74037497357810135460719075010, 6.75557946173021316806256849168, 7.42814923511835420874764519567, 8.933617499905635007068164653641, 9.768259699816330321164151765044, 10.01737908528731549939561120849

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