Properties

Label 2-1472-1.1-c1-0-14
Degree $2$
Conductor $1472$
Sign $1$
Analytic cond. $11.7539$
Root an. cond. $3.42840$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·5-s + 4·7-s + 6·9-s + 2·11-s + 5·13-s − 6·15-s + 4·17-s − 2·19-s − 12·21-s − 23-s − 25-s − 9·27-s + 7·29-s + 3·31-s − 6·33-s + 8·35-s − 2·37-s − 15·39-s − 9·41-s − 8·43-s + 12·45-s − 9·47-s + 9·49-s − 12·51-s − 2·53-s + 4·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s + 1.51·7-s + 2·9-s + 0.603·11-s + 1.38·13-s − 1.54·15-s + 0.970·17-s − 0.458·19-s − 2.61·21-s − 0.208·23-s − 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.538·31-s − 1.04·33-s + 1.35·35-s − 0.328·37-s − 2.40·39-s − 1.40·41-s − 1.21·43-s + 1.78·45-s − 1.31·47-s + 9/7·49-s − 1.68·51-s − 0.274·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $1$
Analytic conductor: \(11.7539\)
Root analytic conductor: \(3.42840\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.553218770\)
\(L(\frac12)\) \(\approx\) \(1.553218770\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−9.866992454756712207798623866216, −8.608238956261300097073169258146, −7.940995723732562677345333914110, −6.56889089382194248049775428165, −6.29658206116634366505284597932, −5.28615499787164158880149394506, −4.88899916461674802073833414510, −3.75737483622489943039302637500, −1.75965596937335757332008532480, −1.11834268237837652771743629830, 1.11834268237837652771743629830, 1.75965596937335757332008532480, 3.75737483622489943039302637500, 4.88899916461674802073833414510, 5.28615499787164158880149394506, 6.29658206116634366505284597932, 6.56889089382194248049775428165, 7.940995723732562677345333914110, 8.608238956261300097073169258146, 9.866992454756712207798623866216

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