Properties

Label 2-220-1.1-c1-0-1
Degree $2$
Conductor $220$
Sign $-1$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  − 2·3-s + 5-s − 4·7-s + 9-s − 11-s − 4·13-s − 2·15-s − 4·19-s + 8·21-s − 6·23-s + 25-s + 4·27-s − 6·29-s + 8·31-s + 2·33-s − 4·35-s + 2·37-s + 8·39-s + 6·41-s + 8·43-s + 45-s + 6·47-s + 9·49-s − 6·53-s − 55-s + 8·57-s − 12·59-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.516·15-s − 0.917·19-s + 1.74·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 1.43·31-s + 0.348·33-s − 0.676·35-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.134·55-s + 1.05·57-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -1)\)

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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−11.99415428568948822443308731111, −10.71041570838860433355440092548, −10.03292455203919149922766850488, −9.151158442340769173810887630430, −7.52651413140030662377321757263, −6.30624098912257478534493446691, −5.84388404672963888992366079306, −4.43743797714188270071922457947, −2.67336507984781548916365505622, 0, 2.67336507984781548916365505622, 4.43743797714188270071922457947, 5.84388404672963888992366079306, 6.30624098912257478534493446691, 7.52651413140030662377321757263, 9.151158442340769173810887630430, 10.03292455203919149922766850488, 10.71041570838860433355440092548, 11.99415428568948822443308731111

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