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  − 2.72·2-s − 1.76·3-s + 5.40·4-s + 0.469·5-s + 4.81·6-s + 1.03·7-s − 9.28·8-s + 0.123·9-s − 1.27·10-s − 1.84·11-s − 9.56·12-s − 0.700·13-s − 2.82·14-s − 0.830·15-s + 14.4·16-s + 0.793·17-s − 0.336·18-s + 3.65·19-s + 2.54·20-s − 1.83·21-s + 5.01·22-s − 3.65·23-s + 16.4·24-s − 4.77·25-s + 1.90·26-s + 5.08·27-s + 5.61·28-s + ⋯
L(s)  = 1  − 1.92·2-s − 1.02·3-s + 2.70·4-s + 0.210·5-s + 1.96·6-s + 0.392·7-s − 3.28·8-s + 0.0411·9-s − 0.404·10-s − 0.555·11-s − 2.75·12-s − 0.194·13-s − 0.755·14-s − 0.214·15-s + 3.61·16-s + 0.192·17-s − 0.0792·18-s + 0.839·19-s + 0.568·20-s − 0.400·21-s + 1.06·22-s − 0.761·23-s + 3.34·24-s − 0.955·25-s + 0.374·26-s + 0.978·27-s + 1.06·28-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 \( 1 + 2.72T + 2T^{2} \)
3 \( 1 + 1.76T + 3T^{2} \)
5 \( 1 - 0.469T + 5T^{2} \)
7 \( 1 - 1.03T + 7T^{2} \)
11 \( 1 + 1.84T + 11T^{2} \)
13 \( 1 + 0.700T + 13T^{2} \)
17 \( 1 - 0.793T + 17T^{2} \)
19 \( 1 - 3.65T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 5.52T + 29T^{2} \)
31 \( 1 - 6.13T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 + 9.69T + 41T^{2} \)
43 \( 1 + 4.25T + 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 + 7.01T + 53T^{2} \)
59 \( 1 - 6.98T + 59T^{2} \)
61 \( 1 + 3.87T + 61T^{2} \)
67 \( 1 + 5.10T + 67T^{2} \)
71 \( 1 - 1.33T + 71T^{2} \)
73 \( 1 + 7.83T + 73T^{2} \)
79 \( 1 - 4.36T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 0.208T + 89T^{2} \)
97 \( 1 - 3.08T + 97T^{2} \)
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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

−10.15375292426754281109691091397, −9.775293799943445489551733263635, −8.462572226034336245936665770895, −7.957444403180484682167740160823, −6.86714877393291162791960352591, −6.08344516614092507177803216195, −5.11360281339546546970207398468, −2.90957299938188625807073647819, −1.50439809214026082281975267437, 0, 1.50439809214026082281975267437, 2.90957299938188625807073647819, 5.11360281339546546970207398468, 6.08344516614092507177803216195, 6.86714877393291162791960352591, 7.957444403180484682167740160823, 8.462572226034336245936665770895, 9.775293799943445489551733263635, 10.15375292426754281109691091397

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