Properties

Label 2-2200-1.1-c1-0-19
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  − 2.56·3-s − 4.56·7-s + 3.56·9-s − 11-s + 1.12·13-s + 7.68·17-s + 1.43·19-s + 11.6·21-s − 1.12·23-s − 1.43·27-s + 8.56·29-s − 1.43·31-s + 2.56·33-s − 7.43·37-s − 2.87·39-s − 12.2·41-s + 3.12·43-s + 11.3·47-s + 13.8·49-s − 19.6·51-s − 9.68·53-s − 3.68·57-s + 1.12·59-s − 12.5·61-s − 16.2·63-s + 2.87·69-s + 3.68·71-s + ⋯
L(s)  = 1  − 1.47·3-s − 1.72·7-s + 1.18·9-s − 0.301·11-s + 0.311·13-s + 1.86·17-s + 0.330·19-s + 2.54·21-s − 0.234·23-s − 0.276·27-s + 1.58·29-s − 0.258·31-s + 0.445·33-s − 1.22·37-s − 0.460·39-s − 1.91·41-s + 0.476·43-s + 1.65·47-s + 1.97·49-s − 2.75·51-s − 1.33·53-s − 0.488·57-s + 0.146·59-s − 1.60·61-s − 2.04·63-s + 0.346·69-s + 0.437·71-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 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 + 4.56T + 7T^{2} \)
13 \( 1 - 1.12T + 13T^{2} \)
17 \( 1 - 7.68T + 17T^{2} \)
19 \( 1 - 1.43T + 19T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 - 8.56T + 29T^{2} \)
31 \( 1 + 1.43T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 3.12T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 9.68T + 53T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 1.12T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 9.68T + 89T^{2} \)
97 \( 1 - 4.87T + 97T^{2} \)
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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

−8.785402291776056964809231708808, −7.67172082986766673362073512417, −6.84440514355326304137414267431, −6.22890009525466020544546426678, −5.64324103213377772433090446760, −4.91771582944466859018289888002, −3.65716234947209555278738095568, −2.95911103601770103329387322429, −1.14086041607664668277828186316, 0, 1.14086041607664668277828186316, 2.95911103601770103329387322429, 3.65716234947209555278738095568, 4.91771582944466859018289888002, 5.64324103213377772433090446760, 6.22890009525466020544546426678, 6.84440514355326304137414267431, 7.67172082986766673362073512417, 8.785402291776056964809231708808

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