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  + 2.50·2-s − 3.08·3-s + 4.25·4-s − 3.57·5-s − 7.72·6-s + 1.44·7-s + 5.64·8-s + 6.53·9-s − 8.95·10-s − 5.34·11-s − 13.1·12-s − 5.39·13-s + 3.60·14-s + 11.0·15-s + 5.60·16-s − 5.31·17-s + 16.3·18-s + 2.56·19-s − 15.2·20-s − 4.45·21-s − 13.3·22-s − 6.63·23-s − 17.4·24-s + 7.81·25-s − 13.4·26-s − 10.9·27-s + 6.13·28-s + ⋯
L(s)  = 1  + 1.76·2-s − 1.78·3-s + 2.12·4-s − 1.60·5-s − 3.15·6-s + 0.545·7-s + 1.99·8-s + 2.17·9-s − 2.83·10-s − 1.61·11-s − 3.79·12-s − 1.49·13-s + 0.964·14-s + 2.85·15-s + 1.40·16-s − 1.28·17-s + 3.85·18-s + 0.588·19-s − 3.40·20-s − 0.971·21-s − 2.85·22-s − 1.38·23-s − 3.55·24-s + 1.56·25-s − 2.64·26-s − 2.09·27-s + 1.16·28-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 \( 1 - 2.50T + 2T^{2} \)
3 \( 1 + 3.08T + 3T^{2} \)
5 \( 1 + 3.57T + 5T^{2} \)
7 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 + 5.34T + 11T^{2} \)
13 \( 1 + 5.39T + 13T^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
19 \( 1 - 2.56T + 19T^{2} \)
23 \( 1 + 6.63T + 23T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 - 4.77T + 31T^{2} \)
37 \( 1 + 3.44T + 37T^{2} \)
41 \( 1 - 8.38T + 41T^{2} \)
43 \( 1 - 5.49T + 43T^{2} \)
47 \( 1 + 0.427T + 47T^{2} \)
53 \( 1 + 3.73T + 53T^{2} \)
59 \( 1 - 2.87T + 59T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 - 6.57T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 0.468T + 73T^{2} \)
79 \( 1 - 5.80T + 79T^{2} \)
83 \( 1 - 0.338T + 83T^{2} \)
89 \( 1 - 1.79T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−10.98140841194093681125593500738, −10.19049290103669260993535737664, −7.896274127225911303889673046168, −7.33781480328545475545579349775, −6.40181130129015525480105856873, −5.29941442933291384467273131701, −4.72547499766496358208227927676, −4.21175914313651092938061032473, −2.57993399509712623727587287083, 0, 2.57993399509712623727587287083, 4.21175914313651092938061032473, 4.72547499766496358208227927676, 5.29941442933291384467273131701, 6.40181130129015525480105856873, 7.33781480328545475545579349775, 7.896274127225911303889673046168, 10.19049290103669260993535737664, 10.98140841194093681125593500738

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