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  − 1.35·2-s + 0.387·3-s − 0.157·4-s − 1.46·5-s − 0.526·6-s + 0.194·7-s + 2.92·8-s − 2.84·9-s + 1.99·10-s + 1.87·11-s − 0.0612·12-s + 6.15·13-s − 0.264·14-s − 0.569·15-s − 3.65·16-s − 5.37·17-s + 3.86·18-s − 3.78·19-s + 0.231·20-s + 0.0755·21-s − 2.54·22-s − 3.43·23-s + 1.13·24-s − 2.84·25-s − 8.35·26-s − 2.26·27-s − 0.0307·28-s + ⋯
L(s)  = 1  − 0.959·2-s + 0.223·3-s − 0.0789·4-s − 0.657·5-s − 0.214·6-s + 0.0735·7-s + 1.03·8-s − 0.949·9-s + 0.630·10-s + 0.564·11-s − 0.0176·12-s + 1.70·13-s − 0.0706·14-s − 0.147·15-s − 0.914·16-s − 1.30·17-s + 0.911·18-s − 0.867·19-s + 0.0518·20-s + 0.0164·21-s − 0.541·22-s − 0.715·23-s + 0.231·24-s − 0.568·25-s − 1.63·26-s − 0.436·27-s − 0.00580·28-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 \( 1 + 1.35T + 2T^{2} \)
3 \( 1 - 0.387T + 3T^{2} \)
5 \( 1 + 1.46T + 5T^{2} \)
7 \( 1 - 0.194T + 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 - 6.15T + 13T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 + 3.78T + 19T^{2} \)
23 \( 1 + 3.43T + 23T^{2} \)
29 \( 1 - 0.0833T + 29T^{2} \)
31 \( 1 + 3.95T + 31T^{2} \)
37 \( 1 - 3.24T + 37T^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 - 2.32T + 43T^{2} \)
47 \( 1 + 8.88T + 47T^{2} \)
53 \( 1 - 7.04T + 53T^{2} \)
59 \( 1 - 7.89T + 59T^{2} \)
61 \( 1 + 2.95T + 61T^{2} \)
67 \( 1 + 1.41T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 0.418T + 73T^{2} \)
79 \( 1 - 0.962T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + 9.44T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−10.31163870373631063687635254628, −9.191196666858178373818624449312, −8.468809337844759310198201518207, −8.231946025707911631362248362465, −6.90727053765502447118163051254, −5.92815962031766334523361713221, −4.41316711357575577296065187600, −3.59325802371127207780347488205, −1.79557396086654244112707139958, 0, 1.79557396086654244112707139958, 3.59325802371127207780347488205, 4.41316711357575577296065187600, 5.92815962031766334523361713221, 6.90727053765502447118163051254, 8.231946025707911631362248362465, 8.468809337844759310198201518207, 9.191196666858178373818624449312, 10.31163870373631063687635254628

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