Properties

Label 2-440-1.1-c1-0-8
Degree $2$
Conductor $440$
Sign $-1$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 3·9-s − 11-s − 8·19-s − 8·23-s + 25-s + 10·29-s + 8·31-s + 2·35-s − 10·37-s − 2·41-s − 6·43-s + 3·45-s − 8·47-s − 3·49-s + 14·53-s + 55-s − 4·59-s + 10·61-s + 6·63-s + 4·67-s − 8·73-s + 2·77-s − 4·79-s + 9·81-s + 10·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 9-s − 0.301·11-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.338·35-s − 1.64·37-s − 0.312·41-s − 0.914·43-s + 0.447·45-s − 1.16·47-s − 3/7·49-s + 1.92·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s + 0.755·63-s + 0.488·67-s − 0.936·73-s + 0.227·77-s − 0.450·79-s + 81-s + 1.09·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−10.52468211895507836187666788110, −9.991595366720817909982952019288, −8.479772891183006960706754270413, −8.327607038382254672069032807798, −6.74350248274942220173363652849, −6.11649847489499863225043030251, −4.79135750379205514131465277722, −3.59974368255131790788583382761, −2.43812931650948031191708956324, 0, 2.43812931650948031191708956324, 3.59974368255131790788583382761, 4.79135750379205514131465277722, 6.11649847489499863225043030251, 6.74350248274942220173363652849, 8.327607038382254672069032807798, 8.479772891183006960706754270413, 9.991595366720817909982952019288, 10.52468211895507836187666788110

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