Properties

Label 2-547-1.1-c1-0-11
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.12·2-s + 1.64·3-s − 0.735·4-s + 3.20·5-s − 1.84·6-s − 0.477·7-s + 3.07·8-s − 0.300·9-s − 3.60·10-s − 2.01·11-s − 1.20·12-s + 4.76·13-s + 0.536·14-s + 5.26·15-s − 1.98·16-s + 5.32·17-s + 0.337·18-s + 3.11·19-s − 2.35·20-s − 0.784·21-s + 2.26·22-s − 8.35·23-s + 5.05·24-s + 5.27·25-s − 5.35·26-s − 5.42·27-s + 0.351·28-s + ⋯
L(s)  = 1  − 0.795·2-s + 0.948·3-s − 0.367·4-s + 1.43·5-s − 0.754·6-s − 0.180·7-s + 1.08·8-s − 0.100·9-s − 1.13·10-s − 0.607·11-s − 0.348·12-s + 1.32·13-s + 0.143·14-s + 1.35·15-s − 0.497·16-s + 1.29·17-s + 0.0795·18-s + 0.715·19-s − 0.527·20-s − 0.171·21-s + 0.482·22-s − 1.74·23-s + 1.03·24-s + 1.05·25-s − 1.05·26-s − 1.04·27-s + 0.0663·28-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

\(L(1)\) \(\approx\) \(1.474011238\)
\(L(\frac12)\) \(\approx\) \(1.474011238\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.12T + 2T^{2} \)
3 \( 1 - 1.64T + 3T^{2} \)
5 \( 1 - 3.20T + 5T^{2} \)
7 \( 1 + 0.477T + 7T^{2} \)
11 \( 1 + 2.01T + 11T^{2} \)
13 \( 1 - 4.76T + 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 - 3.11T + 19T^{2} \)
23 \( 1 + 8.35T + 23T^{2} \)
29 \( 1 - 3.14T + 29T^{2} \)
31 \( 1 + 0.615T + 31T^{2} \)
37 \( 1 - 8.45T + 37T^{2} \)
41 \( 1 - 9.49T + 41T^{2} \)
43 \( 1 + 6.62T + 43T^{2} \)
47 \( 1 - 9.76T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 7.86T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + 6.65T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 9.15T + 79T^{2} \)
83 \( 1 + 8.75T + 83T^{2} \)
89 \( 1 - 17.8T + 89T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21279742051422510141997005073, −9.831936623617115227397751390886, −9.120938409099889642949108439660, −8.252509981266536696933405781957, −7.71996714055407512495463070550, −6.12657836382119095863633503913, −5.43849456537974848942931966441, −3.86540515477316040271816725790, −2.61809558359364754313327321479, −1.35785509045111895414181894750, 1.35785509045111895414181894750, 2.61809558359364754313327321479, 3.86540515477316040271816725790, 5.43849456537974848942931966441, 6.12657836382119095863633503913, 7.71996714055407512495463070550, 8.252509981266536696933405781957, 9.120938409099889642949108439660, 9.831936623617115227397751390886, 10.21279742051422510141997005073

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