Properties

Label 2-546-1.1-c1-0-12
Degree $2$
Conductor $546$
Sign $-1$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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-s + 3-s + 4-s − 2·5-s − 6-s − 7-s − 8-s + 9-s + 2·10-s − 4·11-s + 12-s + 13-s + 14-s − 2·15-s + 16-s − 2·17-s − 18-s − 4·19-s − 2·20-s − 21-s + 4·22-s − 4·23-s − 24-s − 25-s − 26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−10.33940918099112086508153275865, −9.407728550276739147993380802554, −8.419058620635545585592668255564, −7.919520600930853256251460201426, −7.05482338122635591122480401032, −5.93037987998780819768001729788, −4.44015518719984939594103992774, −3.34291279159901837009808602924, −2.15328177975657163080119509125, 0, 2.15328177975657163080119509125, 3.34291279159901837009808602924, 4.44015518719984939594103992774, 5.93037987998780819768001729788, 7.05482338122635591122480401032, 7.919520600930853256251460201426, 8.419058620635545585592668255564, 9.407728550276739147993380802554, 10.33940918099112086508153275865

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