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Properties

Label 2-440-440.109-c0-0-7
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

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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

\(L(\frac{1}{2})\) \(\approx\) \(1.542321455\)
\(L(\frac12)\) \(\approx\) \(1.542321455\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.49224344932933597184742166535, −10.64746265284260230193195696558, −9.526113001601595603269014903391, −8.410931813833626754146247747806, −7.60277852316471105448457529630, −6.77826121012606742620202192372, −5.51387184979870445819446952534, −4.27400744398550312427954188179, −3.27026414301545688130900697464, −2.60574323356518423165035776118, 2.60574323356518423165035776118, 3.27026414301545688130900697464, 4.27400744398550312427954188179, 5.51387184979870445819446952534, 6.77826121012606742620202192372, 7.60277852316471105448457529630, 8.410931813833626754146247747806, 9.526113001601595603269014903391, 10.64746265284260230193195696558, 11.49224344932933597184742166535

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