Properties

Label 2-2200-1.1-c1-0-46
Degree $2$
Conductor $2200$
Sign $-1$
Analytic cond. $17.5670$
Root an. cond. $4.19131$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.67·3-s − 4.38·7-s + 4.16·9-s + 11-s − 4·13-s − 5.87·17-s − 0.526·19-s − 11.7·21-s − 6.15·23-s + 3.11·27-s − 0.967·29-s − 9.60·31-s + 2.67·33-s − 1.76·37-s − 10.7·39-s + 2.50·41-s + 3.85·43-s + 4.91·47-s + 12.2·49-s − 15.7·51-s + 5.29·53-s − 1.40·57-s − 2.63·59-s + 8.96·61-s − 18.2·63-s + 7.97·67-s − 16.4·69-s + ⋯
L(s)  = 1  + 1.54·3-s − 1.65·7-s + 1.38·9-s + 0.301·11-s − 1.10·13-s − 1.42·17-s − 0.120·19-s − 2.56·21-s − 1.28·23-s + 0.600·27-s − 0.179·29-s − 1.72·31-s + 0.465·33-s − 0.290·37-s − 1.71·39-s + 0.391·41-s + 0.588·43-s + 0.716·47-s + 1.74·49-s − 2.20·51-s + 0.727·53-s − 0.186·57-s − 0.343·59-s + 1.14·61-s − 2.30·63-s + 0.974·67-s − 1.98·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(17.5670\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - 2.67T + 3T^{2} \)
7 \( 1 + 4.38T + 7T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 + 0.526T + 19T^{2} \)
23 \( 1 + 6.15T + 23T^{2} \)
29 \( 1 + 0.967T + 29T^{2} \)
31 \( 1 + 9.60T + 31T^{2} \)
37 \( 1 + 1.76T + 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 3.85T + 43T^{2} \)
47 \( 1 - 4.91T + 47T^{2} \)
53 \( 1 - 5.29T + 53T^{2} \)
59 \( 1 + 2.63T + 59T^{2} \)
61 \( 1 - 8.96T + 61T^{2} \)
67 \( 1 - 7.97T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 - 7.61T + 83T^{2} \)
89 \( 1 - 2.83T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−8.962449175860231032389017669762, −7.930931403178202656589240089272, −7.15955575404601703929079540609, −6.60362038565822703757732471735, −5.56505090267812574780012507775, −4.15091864461663081840018832570, −3.69171221823000735682311820468, −2.64698260416310719369938513572, −2.11271391215941515826837604719, 0, 2.11271391215941515826837604719, 2.64698260416310719369938513572, 3.69171221823000735682311820468, 4.15091864461663081840018832570, 5.56505090267812574780012507775, 6.60362038565822703757732471735, 7.15955575404601703929079540609, 7.930931403178202656589240089272, 8.962449175860231032389017669762

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