↑

Properties

Label 2-544-1.1-c1-0-13
Degree $2$
Conductor $544$
Sign $-1$
Analytic cond. $4.34386$
Root an. cond. $2.08419$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 3·9-s − 4·11-s + 2·13-s − 17-s − 4·19-s − 6·23-s − 5·25-s + 8·29-s + 2·31-s + 4·37-s − 2·41-s + 4·43-s − 12·47-s − 3·49-s − 6·53-s − 4·59-s + 4·61-s + 6·63-s − 4·67-s + 6·71-s − 6·73-s + 8·77-s + 10·79-s + 9·81-s + 12·83-s − 10·89-s + ⋯
L(s)  = 1  − 0.755·7-s − 9-s − 1.20·11-s + 0.554·13-s − 0.242·17-s − 0.917·19-s − 1.25·23-s − 25-s + 1.48·29-s + 0.359·31-s + 0.657·37-s − 0.312·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.520·59-s + 0.512·61-s + 0.755·63-s − 0.488·67-s + 0.712·71-s − 0.702·73-s + 0.911·77-s + 1.12·79-s + 81-s + 1.31·83-s − 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $-1$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
17 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−10.35531999633558213630115480527, −9.588152845179069507941515845822, −8.407778739743522637911712748765, −7.949465060430367136751257620846, −6.45486383530876632990023364782, −5.92602088890819392149162970339, −4.68811629121503471986666453564, −3.37777254090361242162912900058, −2.34190413882648615913196631793, 0, 2.34190413882648615913196631793, 3.37777254090361242162912900058, 4.68811629121503471986666453564, 5.92602088890819392149162970339, 6.45486383530876632990023364782, 7.949465060430367136751257620846, 8.407778739743522637911712748765, 9.588152845179069507941515845822, 10.35531999633558213630115480527

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