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  + 2.24·2-s − 0.790·3-s + 3.06·4-s − 3.96·5-s − 1.77·6-s − 4.97·7-s + 2.38·8-s − 2.37·9-s − 8.93·10-s + 6.10·11-s − 2.42·12-s − 0.944·13-s − 11.1·14-s + 3.13·15-s − 0.753·16-s + 0.884·17-s − 5.34·18-s − 2.59·19-s − 12.1·20-s + 3.93·21-s + 13.7·22-s − 2.77·23-s − 1.88·24-s + 10.7·25-s − 2.12·26-s + 4.25·27-s − 15.2·28-s + ⋯
L(s)  = 1  + 1.59·2-s − 0.456·3-s + 1.53·4-s − 1.77·5-s − 0.726·6-s − 1.88·7-s + 0.843·8-s − 0.791·9-s − 2.82·10-s + 1.84·11-s − 0.698·12-s − 0.262·13-s − 2.99·14-s + 0.810·15-s − 0.188·16-s + 0.214·17-s − 1.25·18-s − 0.596·19-s − 2.71·20-s + 0.858·21-s + 2.92·22-s − 0.577·23-s − 0.385·24-s + 2.15·25-s − 0.416·26-s + 0.818·27-s − 2.87·28-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 \( 1 - 2.24T + 2T^{2} \)
3 \( 1 + 0.790T + 3T^{2} \)
5 \( 1 + 3.96T + 5T^{2} \)
7 \( 1 + 4.97T + 7T^{2} \)
11 \( 1 - 6.10T + 11T^{2} \)
13 \( 1 + 0.944T + 13T^{2} \)
17 \( 1 - 0.884T + 17T^{2} \)
19 \( 1 + 2.59T + 19T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 + 1.93T + 29T^{2} \)
31 \( 1 + 3.39T + 31T^{2} \)
37 \( 1 - 2.71T + 37T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 2.84T + 47T^{2} \)
53 \( 1 - 6.58T + 53T^{2} \)
59 \( 1 - 0.838T + 59T^{2} \)
61 \( 1 + 9.71T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 7.51T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 0.398T + 79T^{2} \)
83 \( 1 - 4.09T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + 3.97T + 97T^{2} \)
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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

−10.89602014285924081079803558867, −9.499613370313582155795982576693, −8.534781314766289633945610235685, −7.05015923542252856252033877159, −6.56051400324832413215570162924, −5.73488530902990453408523817742, −4.31967532183064974295782030991, −3.71119906215551841395019411209, −3.03824215480465189659466127013, 0, 3.03824215480465189659466127013, 3.71119906215551841395019411209, 4.31967532183064974295782030991, 5.73488530902990453408523817742, 6.56051400324832413215570162924, 7.05015923542252856252033877159, 8.534781314766289633945610235685, 9.499613370313582155795982576693, 10.89602014285924081079803558867

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