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  − 1.98·2-s − 0.150·3-s + 1.93·4-s + 2.87·5-s + 0.297·6-s − 2.68·7-s + 0.124·8-s − 2.97·9-s − 5.71·10-s + 0.368·11-s − 0.290·12-s − 2.43·13-s + 5.32·14-s − 0.432·15-s − 4.12·16-s − 3.34·17-s + 5.90·18-s + 0.377·19-s + 5.57·20-s + 0.403·21-s − 0.730·22-s − 0.915·23-s − 0.0186·24-s + 3.28·25-s + 4.83·26-s + 0.897·27-s − 5.20·28-s + ⋯
L(s)  = 1  − 1.40·2-s − 0.0866·3-s + 0.968·4-s + 1.28·5-s + 0.121·6-s − 1.01·7-s + 0.0438·8-s − 0.992·9-s − 1.80·10-s + 0.110·11-s − 0.0839·12-s − 0.676·13-s + 1.42·14-s − 0.111·15-s − 1.03·16-s − 0.811·17-s + 1.39·18-s + 0.0866·19-s + 1.24·20-s + 0.0879·21-s − 0.155·22-s − 0.190·23-s − 0.00380·24-s + 0.657·25-s + 0.948·26-s + 0.172·27-s − 0.982·28-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 \( 1 + 1.98T + 2T^{2} \)
3 \( 1 + 0.150T + 3T^{2} \)
5 \( 1 - 2.87T + 5T^{2} \)
7 \( 1 + 2.68T + 7T^{2} \)
11 \( 1 - 0.368T + 11T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 - 0.377T + 19T^{2} \)
23 \( 1 + 0.915T + 23T^{2} \)
29 \( 1 + 2.06T + 29T^{2} \)
31 \( 1 - 5.88T + 31T^{2} \)
37 \( 1 + 7.95T + 37T^{2} \)
41 \( 1 + 1.24T + 41T^{2} \)
43 \( 1 - 5.58T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 1.64T + 61T^{2} \)
67 \( 1 - 5.77T + 67T^{2} \)
71 \( 1 + 7.47T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 0.641T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 - 4.77T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−10.07266297130011687568116441542, −9.444952104738855267235277160253, −8.927168111408678973448682955011, −7.903134333279320218140783763118, −6.69363422728067934765609913970, −6.12866143400486695929947131990, −4.89035369286801028895846755251, −2.97552901282983377560399798346, −1.87513224418067494690345311532, 0, 1.87513224418067494690345311532, 2.97552901282983377560399798346, 4.89035369286801028895846755251, 6.12866143400486695929947131990, 6.69363422728067934765609913970, 7.903134333279320218140783763118, 8.927168111408678973448682955011, 9.444952104738855267235277160253, 10.07266297130011687568116441542

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