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  + 1.87·2-s − 1.13·3-s + 1.52·4-s − 1.30·5-s − 2.13·6-s − 1.71·7-s − 0.896·8-s − 1.70·9-s − 2.45·10-s − 5.30·11-s − 1.73·12-s + 2.18·13-s − 3.21·14-s + 1.48·15-s − 4.72·16-s + 0.392·17-s − 3.19·18-s + 0.498·19-s − 1.98·20-s + 1.95·21-s − 9.96·22-s + 8.33·23-s + 1.02·24-s − 3.29·25-s + 4.10·26-s + 5.35·27-s − 2.60·28-s + ⋯
L(s)  = 1  + 1.32·2-s − 0.657·3-s + 0.761·4-s − 0.584·5-s − 0.872·6-s − 0.647·7-s − 0.317·8-s − 0.567·9-s − 0.775·10-s − 1.60·11-s − 0.500·12-s + 0.606·13-s − 0.859·14-s + 0.384·15-s − 1.18·16-s + 0.0952·17-s − 0.752·18-s + 0.114·19-s − 0.444·20-s + 0.426·21-s − 2.12·22-s + 1.73·23-s + 0.208·24-s − 0.658·25-s + 0.805·26-s + 1.03·27-s − 0.493·28-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 \( 1 - 1.87T + 2T^{2} \)
3 \( 1 + 1.13T + 3T^{2} \)
5 \( 1 + 1.30T + 5T^{2} \)
7 \( 1 + 1.71T + 7T^{2} \)
11 \( 1 + 5.30T + 11T^{2} \)
13 \( 1 - 2.18T + 13T^{2} \)
17 \( 1 - 0.392T + 17T^{2} \)
19 \( 1 - 0.498T + 19T^{2} \)
23 \( 1 - 8.33T + 23T^{2} \)
29 \( 1 + 4.50T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 - 3.06T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 9.93T + 43T^{2} \)
47 \( 1 - 0.714T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 + 0.889T + 61T^{2} \)
67 \( 1 - 4.67T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 5.35T + 73T^{2} \)
79 \( 1 + 7.79T + 79T^{2} \)
83 \( 1 + 17.9T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−10.97630689298517003682423534698, −9.575734136473786346108897130110, −8.487075020995987980102283376711, −7.38917763400977044886838113328, −6.30852546818558768999873208670, −5.51395027032838125160447244813, −4.86469439691153775526129778585, −3.56937138503643972647265578643, −2.79172142806700074040363291214, 0, 2.79172142806700074040363291214, 3.56937138503643972647265578643, 4.86469439691153775526129778585, 5.51395027032838125160447244813, 6.30852546818558768999873208670, 7.38917763400977044886838113328, 8.487075020995987980102283376711, 9.575734136473786346108897130110, 10.97630689298517003682423534698

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