Properties

Label 2-27456-1.1-c1-0-11
Degree $2$
Conductor $27456$
Sign $1$
Analytic cond. $219.237$
Root an. cond. $14.8066$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s + 11-s − 13-s − 2·15-s − 6·17-s + 4·19-s − 8·23-s − 25-s − 27-s + 10·29-s − 33-s − 6·37-s + 39-s + 10·41-s − 4·43-s + 2·45-s + 8·47-s − 7·49-s + 6·51-s + 10·53-s + 2·55-s − 4·57-s + 12·59-s − 14·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.174·33-s − 0.986·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.840·51-s + 1.37·53-s + 0.269·55-s − 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27456\)    =    \(2^{6} \cdot 3 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(219.237\)
Root analytic conductor: \(14.8066\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 27456,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.827260165\)
\(L(\frac12)\) \(\approx\) \(1.827260165\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−15.49800447133753, −14.50783117053532, −14.17908335654713, −13.51619575883578, −13.36952096265459, −12.40628560502106, −12.04900775672564, −11.60978857171258, −10.81219427056809, −10.43831787413497, −9.729474150804866, −9.512652991724287, −8.678043459357197, −8.176977636476881, −7.361781113889509, −6.725764869751797, −6.322531364791525, −5.669110781037255, −5.206106599215582, −4.361414791045512, −3.971023798320213, −2.836625904733328, −2.218630383868047, −1.521837176153214, −0.5456529863658735, 0.5456529863658735, 1.521837176153214, 2.218630383868047, 2.836625904733328, 3.971023798320213, 4.361414791045512, 5.206106599215582, 5.669110781037255, 6.322531364791525, 6.725764869751797, 7.361781113889509, 8.176977636476881, 8.678043459357197, 9.512652991724287, 9.729474150804866, 10.43831787413497, 10.81219427056809, 11.60978857171258, 12.04900775672564, 12.40628560502106, 13.36952096265459, 13.51619575883578, 14.17908335654713, 14.50783117053532, 15.49800447133753

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