Properties

Label 2-2200-1.1-c1-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.59·3-s − 2.72·7-s + 3.72·9-s + 11-s − 13-s + 4.59·17-s + 8.18·19-s − 7.05·21-s − 0.407·23-s + 1.87·27-s + 7.46·29-s − 4.44·31-s + 2.59·33-s + 7.31·37-s − 2.59·39-s − 3.31·41-s + 7.49·43-s − 7.05·47-s + 0.407·49-s + 11.9·51-s − 0.979·53-s + 21.2·57-s + 7.05·59-s − 4.46·61-s − 10.1·63-s + 2.25·67-s − 1.05·69-s + ⋯
L(s)  = 1  + 1.49·3-s − 1.02·7-s + 1.24·9-s + 0.301·11-s − 0.277·13-s + 1.11·17-s + 1.87·19-s − 1.53·21-s − 0.0849·23-s + 0.360·27-s + 1.38·29-s − 0.798·31-s + 0.451·33-s + 1.20·37-s − 0.415·39-s − 0.517·41-s + 1.14·43-s − 1.02·47-s + 0.0581·49-s + 1.66·51-s − 0.134·53-s + 2.81·57-s + 0.918·59-s − 0.571·61-s − 1.27·63-s + 0.275·67-s − 0.127·69-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: \(0\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.959172630\)
\(L(\frac12)\) \(\approx\) \(2.959172630\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.59T + 3T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 4.59T + 17T^{2} \)
19 \( 1 - 8.18T + 19T^{2} \)
23 \( 1 + 0.407T + 23T^{2} \)
29 \( 1 - 7.46T + 29T^{2} \)
31 \( 1 + 4.44T + 31T^{2} \)
37 \( 1 - 7.31T + 37T^{2} \)
41 \( 1 + 3.31T + 41T^{2} \)
43 \( 1 - 7.49T + 43T^{2} \)
47 \( 1 + 7.05T + 47T^{2} \)
53 \( 1 + 0.979T + 53T^{2} \)
59 \( 1 - 7.05T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
67 \( 1 - 2.25T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 3.14T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 - 8.27T + 89T^{2} \)
97 \( 1 + 3.03T + 97T^{2} \)
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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

−9.185444751423294927073492182245, −8.247010027991310129210719166028, −7.64792962566346164825674541541, −6.95873998451486382912092310278, −5.99105333677367314401071032042, −4.98767223769086838781376570756, −3.73733020026033455324557056640, −3.23544908485425440822844515493, −2.50564410769329498266939232607, −1.11458365273105920021744963561, 1.11458365273105920021744963561, 2.50564410769329498266939232607, 3.23544908485425440822844515493, 3.73733020026033455324557056640, 4.98767223769086838781376570756, 5.99105333677367314401071032042, 6.95873998451486382912092310278, 7.64792962566346164825674541541, 8.247010027991310129210719166028, 9.185444751423294927073492182245

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