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  − 0.763·2-s + 1.83·3-s − 1.41·4-s − 1.51·5-s − 1.40·6-s + 1.20·7-s + 2.60·8-s + 0.368·9-s + 1.15·10-s − 5.83·11-s − 2.60·12-s − 5.40·13-s − 0.917·14-s − 2.77·15-s + 0.842·16-s + 1.92·17-s − 0.281·18-s + 0.965·19-s + 2.14·20-s + 2.20·21-s + 4.45·22-s − 0.470·23-s + 4.78·24-s − 2.71·25-s + 4.12·26-s − 4.83·27-s − 1.70·28-s + ⋯
L(s)  = 1  − 0.539·2-s + 1.05·3-s − 0.708·4-s − 0.676·5-s − 0.572·6-s + 0.454·7-s + 0.922·8-s + 0.122·9-s + 0.365·10-s − 1.76·11-s − 0.750·12-s − 1.49·13-s − 0.245·14-s − 0.716·15-s + 0.210·16-s + 0.467·17-s − 0.0662·18-s + 0.221·19-s + 0.479·20-s + 0.481·21-s + 0.950·22-s − 0.0980·23-s + 0.977·24-s − 0.542·25-s + 0.809·26-s − 0.929·27-s − 0.321·28-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 \( 1 + 0.763T + 2T^{2} \)
3 \( 1 - 1.83T + 3T^{2} \)
5 \( 1 + 1.51T + 5T^{2} \)
7 \( 1 - 1.20T + 7T^{2} \)
11 \( 1 + 5.83T + 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 - 0.965T + 19T^{2} \)
23 \( 1 + 0.470T + 23T^{2} \)
29 \( 1 + 3.67T + 29T^{2} \)
31 \( 1 - 0.675T + 31T^{2} \)
37 \( 1 - 5.66T + 37T^{2} \)
41 \( 1 + 5.00T + 41T^{2} \)
43 \( 1 - 3.18T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 8.25T + 53T^{2} \)
59 \( 1 + 5.23T + 59T^{2} \)
61 \( 1 + 3.53T + 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 - 9.68T + 73T^{2} \)
79 \( 1 + 5.23T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + 6.07T + 97T^{2} \)
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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

−10.01758180056237912887182314074, −9.492413967050380411285936085322, −8.376968857332088930930950814179, −7.82136320261286822248046630618, −7.50258150953308367963073788082, −5.40558809003315011742037079781, −4.63429394829218579832683523392, −3.37446911163191721866796934365, −2.23095287451807266078620746413, 0, 2.23095287451807266078620746413, 3.37446911163191721866796934365, 4.63429394829218579832683523392, 5.40558809003315011742037079781, 7.50258150953308367963073788082, 7.82136320261286822248046630618, 8.376968857332088930930950814179, 9.492413967050380411285936085322, 10.01758180056237912887182314074

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