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  − 2.59·2-s + 2.09·3-s + 4.75·4-s − 3.02·5-s − 5.44·6-s − 0.561·7-s − 7.17·8-s + 1.38·9-s + 7.85·10-s + 4.23·11-s + 9.96·12-s − 4.87·13-s + 1.45·14-s − 6.33·15-s + 9.12·16-s − 6.39·17-s − 3.60·18-s + 6.29·19-s − 14.3·20-s − 1.17·21-s − 10.9·22-s − 9.21·23-s − 15.0·24-s + 4.13·25-s + 12.6·26-s − 3.37·27-s − 2.67·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 1.20·3-s + 2.37·4-s − 1.35·5-s − 2.22·6-s − 0.212·7-s − 2.53·8-s + 0.462·9-s + 2.48·10-s + 1.27·11-s + 2.87·12-s − 1.35·13-s + 0.390·14-s − 1.63·15-s + 2.28·16-s − 1.55·17-s − 0.850·18-s + 1.44·19-s − 3.21·20-s − 0.256·21-s − 2.34·22-s − 1.92·23-s − 3.06·24-s + 0.827·25-s + 2.48·26-s − 0.649·27-s − 0.504·28-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 \( 1 + 2.59T + 2T^{2} \)
3 \( 1 - 2.09T + 3T^{2} \)
5 \( 1 + 3.02T + 5T^{2} \)
7 \( 1 + 0.561T + 7T^{2} \)
11 \( 1 - 4.23T + 11T^{2} \)
13 \( 1 + 4.87T + 13T^{2} \)
17 \( 1 + 6.39T + 17T^{2} \)
19 \( 1 - 6.29T + 19T^{2} \)
23 \( 1 + 9.21T + 23T^{2} \)
29 \( 1 - 1.14T + 29T^{2} \)
31 \( 1 + 7.52T + 31T^{2} \)
37 \( 1 - 8.74T + 37T^{2} \)
41 \( 1 - 2.81T + 41T^{2} \)
43 \( 1 + 1.57T + 43T^{2} \)
47 \( 1 + 4.08T + 47T^{2} \)
53 \( 1 + 4.00T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 0.473T + 61T^{2} \)
67 \( 1 - 1.80T + 67T^{2} \)
71 \( 1 + 1.76T + 71T^{2} \)
73 \( 1 - 0.611T + 73T^{2} \)
79 \( 1 + 2.49T + 79T^{2} \)
83 \( 1 - 8.29T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 7.36T + 97T^{2} \)
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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

−9.796166812587156100633519476369, −9.396780810399117537433914868591, −8.640102991994114300422135320980, −7.75638552375296588600312977709, −7.45687439028094087940947598551, −6.39797918579590970291747243451, −4.20694160453719964937338947382, −3.09905397639925749868674914954, −1.93281725158156473046768149933, 0, 1.93281725158156473046768149933, 3.09905397639925749868674914954, 4.20694160453719964937338947382, 6.39797918579590970291747243451, 7.45687439028094087940947598551, 7.75638552375296588600312977709, 8.640102991994114300422135320980, 9.396780810399117537433914868591, 9.796166812587156100633519476369

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