↑

Properties

Label 2-440-440.109-c0-0-3
Degree $2$
Conductor $440$
Sign $1$
Analytic cond. $0.219588$
Root an. cond. $0.468602$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 10-s + 11-s − 12-s − 14-s − 15-s + 16-s − 17-s − 19-s + 20-s + 21-s + 22-s − 24-s + 25-s + 27-s − 28-s − 29-s − 30-s − 31-s + 32-s − 33-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 10-s + 11-s − 12-s − 14-s − 15-s + 16-s − 17-s − 19-s + 20-s + 21-s + 22-s − 24-s + 25-s + 27-s − 28-s − 29-s − 30-s − 31-s + 32-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.219588\)
Root analytic conductor: \(0.468602\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{440} (109, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.203590666\)
\(L(\frac12)\) \(\approx\) \(1.203590666\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.30760242304718401420675634499, −10.80237501820045346716064373016, −9.769061055061837628316273718683, −8.808518338963401595855599801284, −6.95124049271711626634931610216, −6.39079666766011395590792057464, −5.79859937462210122165599987861, −4.75471478260800469118493250092, −3.48796811994734313649172580572, −2.01548309327795849820138984267, 2.01548309327795849820138984267, 3.48796811994734313649172580572, 4.75471478260800469118493250092, 5.79859937462210122165599987861, 6.39079666766011395590792057464, 6.95124049271711626634931610216, 8.808518338963401595855599801284, 9.769061055061837628316273718683, 10.80237501820045346716064373016, 11.30760242304718401420675634499

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