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  − 1.74·2-s − 1.27·3-s + 1.04·4-s − 3.61·5-s + 2.22·6-s + 4.28·7-s + 1.66·8-s − 1.37·9-s + 6.30·10-s − 1.82·11-s − 1.32·12-s + 1.54·13-s − 7.46·14-s + 4.60·15-s − 4.99·16-s + 5.89·17-s + 2.40·18-s + 5.77·19-s − 3.77·20-s − 5.44·21-s + 3.17·22-s − 7.36·23-s − 2.12·24-s + 8.06·25-s − 2.70·26-s + 5.57·27-s + 4.47·28-s + ⋯
L(s)  = 1  − 1.23·2-s − 0.734·3-s + 0.522·4-s − 1.61·5-s + 0.906·6-s + 1.61·7-s + 0.589·8-s − 0.459·9-s + 1.99·10-s − 0.549·11-s − 0.383·12-s + 0.429·13-s − 1.99·14-s + 1.18·15-s − 1.24·16-s + 1.43·17-s + 0.567·18-s + 1.32·19-s − 0.844·20-s − 1.18·21-s + 0.677·22-s − 1.53·23-s − 0.433·24-s + 1.61·25-s − 0.529·26-s + 1.07·27-s + 0.845·28-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 \( 1 + 1.74T + 2T^{2} \)
3 \( 1 + 1.27T + 3T^{2} \)
5 \( 1 + 3.61T + 5T^{2} \)
7 \( 1 - 4.28T + 7T^{2} \)
11 \( 1 + 1.82T + 11T^{2} \)
13 \( 1 - 1.54T + 13T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 - 5.77T + 19T^{2} \)
23 \( 1 + 7.36T + 23T^{2} \)
29 \( 1 + 4.20T + 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 + 9.26T + 37T^{2} \)
41 \( 1 - 8.29T + 41T^{2} \)
43 \( 1 + 2.50T + 43T^{2} \)
47 \( 1 + 6.07T + 47T^{2} \)
53 \( 1 + 4.35T + 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 + 9.67T + 61T^{2} \)
67 \( 1 + 3.67T + 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 - 2.36T + 73T^{2} \)
79 \( 1 - 6.27T + 79T^{2} \)
83 \( 1 + 2.57T + 83T^{2} \)
89 \( 1 - 8.45T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−10.70825437421141646532254558436, −9.377013137894505330355511850118, −8.259085006755731849916043186721, −7.87036903374789733595981187781, −7.38128213414676656785782011637, −5.59291010373644259048661406225, −4.79708462732473814585122762878, −3.57376572789881180246185263803, −1.43478959487317990534492372936, 0, 1.43478959487317990534492372936, 3.57376572789881180246185263803, 4.79708462732473814585122762878, 5.59291010373644259048661406225, 7.38128213414676656785782011637, 7.87036903374789733595981187781, 8.259085006755731849916043186721, 9.377013137894505330355511850118, 10.70825437421141646532254558436

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