Properties

Label 2-547-1.1-c1-0-30
Degree $2$
Conductor $547$
Sign $1$
Analytic cond. $4.36781$
Root an. cond. $2.08993$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.98·2-s + 2.42·3-s + 1.93·4-s − 0.826·5-s + 4.80·6-s + 0.383·7-s − 0.136·8-s + 2.86·9-s − 1.63·10-s + 3.48·11-s + 4.67·12-s − 0.283·13-s + 0.760·14-s − 2.00·15-s − 4.13·16-s − 4.93·17-s + 5.68·18-s − 1.37·19-s − 1.59·20-s + 0.929·21-s + 6.91·22-s + 4.28·23-s − 0.331·24-s − 4.31·25-s − 0.561·26-s − 0.317·27-s + 0.740·28-s + ⋯
L(s)  = 1  + 1.40·2-s + 1.39·3-s + 0.965·4-s − 0.369·5-s + 1.96·6-s + 0.144·7-s − 0.0483·8-s + 0.956·9-s − 0.518·10-s + 1.05·11-s + 1.35·12-s − 0.0785·13-s + 0.203·14-s − 0.516·15-s − 1.03·16-s − 1.19·17-s + 1.34·18-s − 0.314·19-s − 0.356·20-s + 0.202·21-s + 1.47·22-s + 0.892·23-s − 0.0676·24-s − 0.863·25-s − 0.110·26-s − 0.0610·27-s + 0.139·28-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: $1$
Analytic conductor: \(4.36781\)
Root analytic conductor: \(2.08993\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 547,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.071807269\)
\(L(\frac12)\) \(\approx\) \(4.071807269\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.98T + 2T^{2} \)
3 \( 1 - 2.42T + 3T^{2} \)
5 \( 1 + 0.826T + 5T^{2} \)
7 \( 1 - 0.383T + 7T^{2} \)
11 \( 1 - 3.48T + 11T^{2} \)
13 \( 1 + 0.283T + 13T^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
19 \( 1 + 1.37T + 19T^{2} \)
23 \( 1 - 4.28T + 23T^{2} \)
29 \( 1 + 2.57T + 29T^{2} \)
31 \( 1 - 8.23T + 31T^{2} \)
37 \( 1 - 1.31T + 37T^{2} \)
41 \( 1 - 1.17T + 41T^{2} \)
43 \( 1 + 2.33T + 43T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 + 6.14T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 3.13T + 61T^{2} \)
67 \( 1 - 5.10T + 67T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 - 3.77T + 73T^{2} \)
79 \( 1 + 2.61T + 79T^{2} \)
83 \( 1 - 4.34T + 83T^{2} \)
89 \( 1 - 2.11T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−11.22141317409874644498095098736, −9.700049726159845866469309897206, −8.932851140487065985312077515794, −8.196974513115901900266989635665, −7.02559640833648567315210605294, −6.20451317394968631824854064001, −4.75636201247294252293608887270, −4.01308399222230810345890542348, −3.18984908893420473839407497653, −2.10975171258152058186983214009, 2.10975171258152058186983214009, 3.18984908893420473839407497653, 4.01308399222230810345890542348, 4.75636201247294252293608887270, 6.20451317394968631824854064001, 7.02559640833648567315210605294, 8.196974513115901900266989635665, 8.932851140487065985312077515794, 9.700049726159845866469309897206, 11.22141317409874644498095098736

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